Algebraic expressions and rational functions with simplified fractions and quadratic equations.

It looks like you've uploaded an image containing a list of rational expressions, possibly asking to simplify or match them. However, the layout suggests it might be a matching exercise: on the left are simplified expressions, and on the right are more complex ones — perhaps the goal is to simplify the expressions on the right and match them with the corresponding simplified forms on the left.

Let’s go through each expression on the right-hand side, simplify it, and see which one matches the expressions on the left-hand side.

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Step 1: Simplify each expression on the right

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#### 1.
$$
\frac{x^2 + 7x + 12}{(x+4)(x-9)}
$$

Factor numerator:
$$
x^2 + 7x + 12 = (x+3)(x+4)
$$

So:
$$
\frac{(x+3)(x+4)}{(x+4)(x-9)} = \frac{x+3}{x-9}, \quad x \ne -4
$$

This does not match any expression on the left directly.

---

#### 2.
$$
\frac{x^2 + 7x + 12}{x^2 - 5x - 36}
$$

Numerator: $x^2 + 7x + 12 = (x+3)(x+4)$

Denominator: $x^2 - 5x - 36 = (x-9)(x+4)$

So:
$$
\frac{(x+3)(x+4)}{(x-9)(x+4)} = \frac{x+3}{x-9}, \quad x \ne -4
$$

Same as above — doesn't match anything on the left.

---

#### 3.
$$
\frac{2x^2 + 14x + 24}{x^2 - 5x - 36}
$$

Numerator: $2x^2 + 14x + 24 = 2(x^2 + 7x + 12) = 2(x+3)(x+4)$

Denominator: $x^2 - 5x - 36 = (x-9)(x+4)$

So:
$$
\frac{2(x+3)(x+4)}{(x-9)(x+4)} = \frac{2(x+3)}{x-9}, \quad x \ne -4
$$

Still not matching left.

---

#### 4.
$$
\frac{2x^2 + 14x + 24}{3x^2 - 15x - 108}
$$

Numerator: $2x^2 + 14x + 24 = 2(x^2 + 7x + 12) = 2(x+3)(x+4)$

Denominator: $3x^2 - 15x - 108 = 3(x^2 - 5x - 36) = 3(x-9)(x+4)$

So:
$$
\frac{2(x+3)(x+4)}{3(x-9)(x+4)} = \frac{2(x+3)}{3(x-9)}, \quad x \ne -4
$$

Still not matching.

---

#### 5.
$$
\frac{2x^2 + 14x + 24}{3x^2 + 4x - 15}
$$

Numerator: same as before → $2(x+3)(x+4)$

Denominator: $3x^2 + 4x - 15$

Try factoring: find two numbers that multiply to $3 \times (-15) = -45$, add to 4 → 9 and -5

So:
$$
3x^2 + 4x - 15 = 3x^2 + 9x - 5x - 15 = 3x(x+3) -5(x+3) = (3x - 5)(x + 3)
$$

So:
$$
\frac{2(x+3)(x+4)}{(3x - 5)(x + 3)} = \frac{2(x+4)}{3x - 5}, \quad x \ne -3
$$

Not matching.

---

#### 6.
$$
\frac{14x - 24 - 2x^2}{3x^2 + 4x - 15}
$$

First, rewrite numerator in standard form:
$$
-2x^2 + 14x - 24 = -2(x^2 - 7x + 12) = -2(x-3)(x-4)
$$

Denominator: already factored as $(3x - 5)(x + 3)$ from earlier

So:
$$
\frac{-2(x-3)(x-4)}{(3x - 5)(x + 3)}
$$

No common factors — can’t simplify further.

Doesn’t match anything.

---

#### 7.
$$
\frac{14x - 24 - 2x^2}{3x^2 - 4x - 15}
$$

Numerator: same as above: $-2x^2 + 14x - 24 = -2(x^2 - 7x + 12) = -2(x-3)(x-4)$

Denominator: $3x^2 - 4x - 15$

Factor: look for two numbers that multiply to $3 \times (-15) = -45$, add to -4 → -9 and 5

So:
$$
3x^2 - 4x - 15 = 3x^2 - 9x + 5x - 15 = 3x(x-3) + 5(x-3) = (3x + 5)(x - 3)
$$

So:
$$
\frac{-2(x-3)(x-4)}{(3x + 5)(x - 3)} = \frac{-2(x-4)}{3x + 5}, \quad x \ne 3
$$

Still not matching.

---

Now let’s look at the left-hand side expressions and see if any of them can be simplified or matched.

---

Left-hand side:

1. $\frac{2(x+3)}{3(x+3)}$ → cancel $(x+3)$ → $\frac{2}{3}$, $x \ne -3$

2. $\frac{2(x+3)}{x+3}$ → $\frac{2}{1} = 2$, $x \ne -3$

3. $\frac{x+3}{2(x+3)}$ → $\frac{1}{2}$, $x \ne -3$

4. $\frac{x+3}{2x+6}$ → $2x+6 = 2(x+3)$, so $\frac{x+3}{2(x+3)} = \frac{1}{2}$, $x \ne -3$

5. $\frac{(x+3)(x+4)}{2x+6}$ → $2x+6 = 2(x+3)$, so $\frac{(x+3)(x+4)}{2(x+3)} = \frac{x+4}{2}$, $x \ne -3$

6. $\frac{(x+3)(x+4)}{2x+8}$ → $2x+8 = 2(x+4)$, so $\frac{(x+3)(x+4)}{2(x+4)} = \frac{x+3}{2}$, $x \ne -4$

7. $\frac{x^2 + 7x + 12}{2x+8}$ → numerator: $(x+3)(x+4)$, denominator: $2(x+4)$, so $\frac{(x+3)(x+4)}{2(x+4)} = \frac{x+3}{2}$, $x \ne -4$

Wait! That's interesting.

Let’s check number 7 on the left:
$$
\frac{x^2 + 7x + 12}{2x+8} = \frac{(x+3)(x+4)}{2(x+4)} = \frac{x+3}{2}, \quad x \ne -4
$$

And earlier, we saw that expression 6 on the right was:
$$
\frac{(x+3)(x+4)}{2x+8} = \frac{x+3}{2}
$$

So this matches!

But wait — both sides have the same expression? Let's double-check.

Actually, looking back:

On the left, the last expression is:
$$
\frac{x^2 + 7x + 12}{2x+8}
$$

On the right, we have:
$$
\frac{14x - 24 - 2x^2}{3x^2 - 4x - 15}
$$

But no — the right column has 7 expressions, and the left column also has 7.

Looking again:

Left column:
1. $\frac{2(x+3)}{3(x+3)}$
2. $\frac{2(x+3)}{x+3}$
3. $\frac{x+3}{2(x+3)}$
4. $\frac{x+3}{2x+6}$
5. $\frac{(x+3)(x+4)}{2x+6}$
6. $\frac{(x+3)(x+4)}{2x+8}$
7. $\frac{x^2 + 7x + 12}{2x+8}$

Right column:
1. $\frac{x^2 + 7x + 12}{(x+4)(x-9)}$
2. $\frac{x^2 + 7x + 12}{x^2 - 5x - 36}$
3. $\frac{2x^2 + 14x + 24}{x^2 - 5x - 36}$
4. $\frac{2x^2 + 14x + 24}{3x^2 - 15x - 108}$
5. $\frac{2x^2 + 14x + 24}{3x^2 + 4x - 15}$
6. $\frac{14x - 24 - 2x^2}{3x^2 + 4x - 15}$
7. $\frac{14x - 24 - 2x^2}{3x^2 - 4x - 15}$

Now let’s try to simplify each right-side expression and see which one matches a left-side one.

We already did several. Let's do #7 on the right:

$$
\frac{14x - 24 - 2x^2}{3x^2 - 4x - 15} = \frac{-2x^2 + 14x - 24}{3x^2 - 4x - 15}
$$

As before:
Numerator: $-2(x^2 - 7x + 12) = -2(x-3)(x-4)$
Denominator: $(3x + 5)(x - 3)$

So:
$$
= \frac{-2(x-3)(x-4)}{(3x+5)(x-3)} = \frac{-2(x-4)}{3x+5}, \quad x \ne 3
$$

Not matching.

Now go back to left #6 and left #7:

They are both:
- $\frac{(x+3)(x+4)}{2x+8}$ → $2x+8 = 2(x+4)$ → simplifies to $\frac{x+3}{2}$
- $\frac{x^2 + 7x + 12}{2x+8}$ → same thing: $(x+3)(x+4)/[2(x+4)] = (x+3)/2$

So left #6 and #7 are equivalent after simplification.

But now, look at right #3:
$$
\frac{2x^2 + 14x + 24}{x^2 - 5x - 36}
$$

We simplified this to:
$$
\frac{2(x+3)(x+4)}{(x-9)(x+4)} = \frac{2(x+3)}{x-9}
$$

Not matching.

But what about right #4:
$$
\frac{2x^2 + 14x + 24}{3x^2 - 15x - 108}
= \frac{2(x+3)(x+4)}{3(x-9)(x+4)} = \frac{2(x+3)}{3(x-9)}
$$

Still not matching.

Wait — look at left #1: $\frac{2(x+3)}{3(x+3)} = \frac{2}{3}$, $x \ne -3$

Is there a right-side expression that simplifies to $\frac{2}{3}$?

Only if numerator and denominator are proportional.

Look at right #4 again:
$$
\frac{2x^2 + 14x + 24}{3x^2 - 15x - 108}
$$

We had:
Numerator: $2(x+3)(x+4)$
Denominator: $3(x-9)(x+4)$

So:
$$
\frac{2(x+3)(x+4)}{3(x-9)(x+4)} = \frac{2(x+3)}{3(x-9)}
$$

Only equals $2/3$ if $x+3 = x-9$, which is impossible.

So not constant.

But now consider left #4: $\frac{x+3}{2x+6} = \frac{x+3}{2(x+3)} = \frac{1}{2}$

Similarly, left #3: $\frac{x+3}{2(x+3)} = \frac{1}{2}$

So multiple expressions simplify to constants.

But none of the right expressions simplify to constants unless they are identical up to cancellation.

Wait — let’s check right #5:
$$
\frac{2x^2 + 14x + 24}{3x^2 + 4x - 15}
= \frac{2(x+3)(x+4)}{(3x - 5)(x + 3)} = \frac{2(x+4)}{3x - 5}
$$

Not constant.

But now — look at left #6: $\frac{(x+3)(x+4)}{2x+8} = \frac{(x+3)(x+4)}{2(x+4)} = \frac{x+3}{2}$

And left #7: $\frac{x^2 + 7x + 12}{2x+8} = \frac{(x+3)(x+4)}{2(x+4)} = \frac{x+3}{2}$

So both are equal to $\frac{x+3}{2}$

Now, is there a right-side expression that simplifies to $\frac{x+3}{2}$?

Yes — but only if it's exactly that.

But all right expressions are more complicated.

Wait — perhaps the task is to simplify each expression on the right and then match it with a simplified form on the left.

But none of the right expressions simplify to a constant or simple linear fraction like those on the left.

Unless we made a mistake.

Wait — look at right #1:
$$
\frac{x^2 + 7x + 12}{(x+4)(x-9)} = \frac{(x+3)(x+4)}{(x+4)(x-9)} = \frac{x+3}{x-9}
$$

Not matching.

But what about right #2:
$$
\frac{x^2 + 7x + 12}{x^2 - 5x - 36} = \frac{(x+3)(x+4)}{(x-9)(x+4)} = \frac{x+3}{x-9}
$$

Same.

Now, look at left #5: $\frac{(x+3)(x+4)}{2x+6} = \frac{(x+3)(x+4)}{2(x+3)} = \frac{x+4}{2}$

So this simplifies to $\frac{x+4}{2}$

Is there a right expression that simplifies to $\frac{x+4}{2}$?

No.

But here's a possibility: maybe the task is to simplify the expressions on the right, and then see which ones are equivalent to the left ones.

Alternatively, perhaps the left side is the simplified form, and the right side is the unsimplified form, and we need to match them.

Let’s try to see if any right-side expression simplifies to one of the left-side ones.

Let’s take left #1: $\frac{2(x+3)}{3(x+3)} = \frac{2}{3}$

Is there a right expression that simplifies to $\frac{2}{3}$? Only if numerator and denominator are proportional.

For example, if we had:
$$
\frac{2(x+3)}{3(x+3)} = \frac{2}{3}
$$

But that’s already on the left.

Now, look at right #4:
$$
\frac{2x^2 + 14x + 24}{3x^2 - 15x - 108}
$$

We had:
$$
= \frac{2(x+3)(x+4)}{3(x-9)(x+4)} = \frac{2(x+3)}{3(x-9)}
$$

This is not equal to $\frac{2}{3}$ unless $x+3 = x-9$, impossible.

But notice: left #4 is $\frac{x+3}{2x+6} = \frac{x+3}{2(x+3)} = \frac{1}{2}$

Similarly, left #3 is $\frac{x+3}{2(x+3)} = \frac{1}{2}$

So these are equivalent.

Now, is there a right expression that simplifies to $\frac{1}{2}$?

Let’s try to see if any right expression simplifies to a constant.

For example, suppose we have:
$$
\frac{2x^2 + 14x + 24}{4x^2 + 28x + 48} = \frac{2(x+3)(x+4)}{4(x+3)(x+4)} = \frac{2}{4} = \frac{1}{2}
$$

But that's not in the list.

None of the right expressions have such a form.

But wait — look at left #6 and left #7 — they are both equal to $\frac{x+3}{2}$

And right #7:
$$
\frac{14x - 24 - 2x^2}{3x^2 - 4x - 15} = \frac{-2x^2 + 14x - 24}{3x^2 - 4x - 15}
$$

We simplified to $\frac{-2(x-4)}{3x+5}$

Not matching.

But perhaps there's a typo or misalignment.

Wait — let’s re-express left #6 and left #7:

- Left #6: $\frac{(x+3)(x+4)}{2x+8} = \frac{(x+3)(x+4)}{2(x+4)} = \frac{x+3}{2}$
- Left #7: $\frac{x^2 + 7x + 12}{2x+8} = \frac{(x+3)(x+4)}{2(x+4)} = \frac{x+3}{2}$

So both are the same.

Now, look at right #3:
$$
\frac{2x^2 + 14x + 24}{x^2 - 5x - 36} = \frac{2(x+3)(x+4)}{(x-9)(x+4)} = \frac{2(x+3)}{x-9}
$$

Not matching.

But here's an idea: maybe the task is to simplify the right-hand expressions and then identify which ones are undefined at certain values, or find their simplified form.

Alternatively, perhaps the image is meant to be paired, and we need to match each right expression with its simplified form on the left.

But none of the right expressions simplify to any of the left ones.

Unless we missed one.

Let’s try right #5:
$$
\frac{2x^2 + 14x + 24}{3x^2 + 4x - 15} = \frac{2(x+3)(x+4)}{(3x - 5)(x + 3)} = \frac{2(x+4)}{3x - 5}
$$

Not matching.

What about right #6:
$$
\frac{14x - 24 - 2x^2}{3x^2 + 4x - 15} = \frac{-2x^2 + 14x - 24}{(3x - 5)(x + 3)} = \frac{-2(x-3)(x-4)}{(3x - 5)(x + 3)}
$$

No common factors.

But now — look at left #5: $\frac{(x+3)(x+4)}{2x+6} = \frac{(x+3)(x+4)}{2(x+3)} = \frac{x+4}{2}$

Is there a right expression that simplifies to $\frac{x+4}{2}$?

No.

But perhaps the task is to simplify each expression on the right, and the answer is to write the simplified form.

But since the user said "solve the problem", and there's no explicit question, I think the intended task is to simplify each rational expression on the right, and match it with the corresponding simplified form on the left.

But none match.

Wait — let’s look at right #4:
$$
\frac{2x^2 + 14x + 24}{3x^2 - 15x - 108}
= \frac{2(x+3)(x+4)}{3(x-9)(x+4)} = \frac{2(x+3)}{3(x-9)}
$$

Now, is there a left expression that is $\frac{2(x+3)}{3(x-9)}$? No.

But left #1 is $\frac{2(x+3)}{3(x+3)} = \frac{2}{3}$

Different.

Perhaps the only match is between left #6 and left #7, which are equivalent.

But they are both on the left.

Maybe the task is to simplify the left expressions first, and then see which ones are equivalent.

Let’s do that.

Simplify left-hand side:

1. $\frac{2(x+3)}{3(x+3)} = \frac{2}{3}$, $x \ne -3$
2. $\frac{2(x+3)}{x+3} = 2$, $x \ne -3$
3. $\frac{x+3}{2(x+3)} = \frac{1}{2}$, $x \ne -3$
4. $\frac{x+3}{2x+6} = \frac{x+3}{2(x+3)} = \frac{1}{2}$, $x \ne -3$
5. $\frac{(x+3)(x+4)}{2x+6} = \frac{(x+3)(x+4)}{2(x+3)} = \frac{x+4}{2}$, $x \ne -3$
6. $\frac{(x+3)(x+4)}{2x+8} = \frac{(x+3)(x+4)}{2(x+4)} = \frac{x+3}{2}$, $x \ne -4$
7. $\frac{x^2 + 7x + 12}{2x+8} = \frac{(x+3)(x+4)}{2(x+4)} = \frac{x+3}{2}$, $x \ne -4$

So simplified forms:
- 1: $2/3$
- 2: 2
- 3: $1/2$
- 4: $1/2$
- 5: $(x+4)/2$
- 6: $(x+3)/2$
- 7: $(x+3)/2$

Now simplify right-hand side:

1. $\frac{x^2 + 7x + 12}{(x+4)(x-9)} = \frac{(x+3)(x+4)}{(x+4)(x-9)} = \frac{x+3}{x-9}$
2. $\frac{x^2 + 7x + 12}{x^2 - 5x - 36} = \frac{(x+3)(x+4)}{(x-9)(x+4)} = \frac{x+3}{x-9}$
3. $\frac{2x^2 + 14x + 24}{x^2 - 5x - 36} = \frac{2(x+3)(x+4)}{(x-9)(x+4)} = \frac{2(x+3)}{x-9}$
4. $\frac{2x^2 + 14x + 24}{3x^2 - 15x - 108} = \frac{2(x+3)(x+4)}{3(x-9)(x+4)} = \frac{2(x+3)}{3(x-9)}$
5. $\frac{2x^2 + 14x + 24}{3x^2 + 4x - 15} = \frac{2(x+3)(x+4)}{(3x - 5)(x + 3)} = \frac{2(x+4)}{3x - 5}$
6. $\frac{14x - 24 - 2x^2}{3x^2 + 4x - 15} = \frac{-2(x-3)(x-4)}{(3x - 5)(x + 3)}$
7. $\frac{14x - 24 - 2x^2}{3x^2 - 4x - 15} = \frac{-2(x-3)(x-4)}{(3x + 5)(x - 3)} = \frac{-2(x-4)}{3x + 5}$

Now, compare with left side.

The only possible match is if any right expression simplifies to one of the left ones.

But none do.

Unless the task is to identify which left expressions are equivalent.

For example:
- Left #3 and #4 both simplify to $1/2$
- Left #6 and #7 both simplify to $(x+3)/2$

So perhaps the solution is to group the equivalent expressions.

Or maybe the task is to simplify each expression.

Given the context, I think the most likely intended task is to simplify each rational expression, and the answer is the simplified form.

Since the user said "solve the problem", and there's no specific instruction, I'll assume the task is to simplify the expressions on the right, and perhaps match them.

But since none match the left, perhaps the left is just a list of examples.

Alternatively, maybe the image is from a worksheet where you are to simplify each expression on the right and write the simplified form.

So let's provide the simplified form of each right-hand expression:

Final Answer:

Simplify each expression on the right:

1. $\frac{x^2 + 7x + 12}{(x+4)(x-9)} = \frac{x+3}{x-9}, \quad x \ne -4$

2. $\frac{x^2 + 7x + 12}{x^2 - 5x - 36} = \frac{x+3}{x-9}, \quad x \ne -4$

3. $\frac{2x^2 + 14x + 24}{x^2 - 5x - 36} = \frac{2(x+3)}{x-9}, \quad x \ne -4$

4. $\frac{2x^2 + 14x + 24}{3x^2 - 15x - 108} = \frac{2(x+3)}{3(x-9)}, \quad x \ne -4$

5. $\frac{2x^2 + 14x + 24}{3x^2 + 4x - 15} = \frac{2(x+4)}{3x - 5}, \quad x \ne -3$

6. $\frac{14x - 24 - 2x^2}{3x^2 + 4x - 15} = \frac{-2(x-3)(x-4)}{(3x - 5)(x + 3)}$

7. $\frac{14x - 24 - 2x^2}{3x^2 - 4x - 15} = \frac{-2(x-4)}{3x + 5}, \quad x \ne 3$

And for the left side, the simplified forms are:

1. $\frac{2}{3}, \quad x \ne -3$
2. $2, \quad x \ne -3$
3. $\frac{1}{2}, \quad x \ne -3$
4. $\frac{1}{2}, \quad x \ne -3$
5. $\frac{x+4}{2}, \quad x \ne -3$
6. $\frac{x+3}{2}, \quad x \ne -4$
7. $\frac{x+3}{2}, \quad x \ne -4$

So the only matching simplified forms are:
- Left #3 and #4 are both $1/2$
- Left #6 and #7 are both $(x+3)/2$

Otherwise, no direct match.

If the task is to simplify, then the above are the answers.

If the task is to match, then there is no perfect match.

But given the structure, perhaps the intended task is to simplify the expressions on the right, and the left side is just distractors or examples.

So the final answer is the simplified forms of the right-hand expressions as shown above.