High school algebra worksheet featuring 12 problems on simplifying rational expressions through multiplication and division.

Problem: Simplify the given expressions. Divide as indicated. Use separate paper if necessary. Factor each expression where possible.

The task involves simplifying algebraic expressions, performing division where indicated, and factoring expressions where possible. Below is a step-by-step solution for each part of the problem.

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#### 1. Simplify:
$$
\frac{x^2 - 18}{x + 3} \quad \text{and} \quad \frac{15x - 54}{x - 2}
$$

##### Step 1.1: Simplify $\frac{x^2 - 18}{x + 3}$
- The numerator $x^2 - 18$ can be factored using the difference of squares:
$$
x^2 - 18 = (x - \sqrt{18})(x + \sqrt{18}) = (x - 3\sqrt{2})(x + 3\sqrt{2})
$$
- However, since $x^2 - 18$ does not factor into $(x + 3)$ or $(x - 3)$, we cannot simplify this fraction further.
- Therefore:
$$
\frac{x^2 - 18}{x + 3} \quad \text{(cannot be simplified further)}
$$

##### Step 1.2: Simplify $\frac{15x - 54}{x - 2}$
- Factor the numerator $15x - 54$:
$$
15x - 54 = 3(5x - 18)
$$
- The denominator is $x - 2$, which does not match any factor in the numerator.
- Therefore:
$$
\frac{15x - 54}{x - 2} = \frac{3(5x - 18)}{x - 2} \quad \text{(cannot be simplified further)}
$$

---

#### 2. Simplify:
$$
\frac{x^2 - 7x - 16}{3x - 9} \quad \text{and} \quad \frac{x^2 + 4x + 5}{x^2 - 4x + 5}
$$

##### Step 2.1: Simplify $\frac{x^2 - 7x - 16}{3x - 9}$
- Factor the numerator $x^2 - 7x - 16$:
$$
x^2 - 7x - 16 = (x - 8)(x + 2)
$$
- Factor the denominator $3x - 9$:
$$
3x - 9 = 3(x - 3)
$$
- The fraction becomes:
$$
\frac{x^2 - 7x - 16}{3x - 9} = \frac{(x - 8)(x + 2)}{3(x - 3)}
$$
- Since there are no common factors between the numerator and the denominator, the expression cannot be simplified further:
$$
\frac{x^2 - 7x - 16}{3x - 9} = \frac{(x - 8)(x + 2)}{3(x - 3)}
$$

##### Step 2.2: Simplify $\frac{x^2 + 4x + 5}{x^2 - 4x + 5}$
- The numerator $x^2 + 4x + 5$ and the denominator $x^2 - 4x + 5$ do not factor further over the real numbers.
- Therefore:
$$
\frac{x^2 + 4x + 5}{x^2 - 4x + 5} \quad \text{(cannot be simplified further)}
$$

---

#### 3. Simplify:
$$
\frac{5x^2 - 40x + 45}{5x} \quad \text{and} \quad \frac{x^2 + 2x - 28}{x^2 - 3x + 2}
$$

##### Step 3.1: Simplify $\frac{5x^2 - 40x + 45}{5x}$
- Factor the numerator $5x^2 - 40x + 45$:
$$
5x^2 - 40x + 45 = 5(x^2 - 8x + 9) = 5(x - 3)(x - 3) = 5(x - 3)^2
$$
- The denominator is $5x$. Divide both the numerator and the denominator by $5$:
$$
\frac{5x^2 - 40x + 45}{5x} = \frac{5(x - 3)^2}{5x} = \frac{(x - 3)^2}{x}
$$
- Therefore:
$$
\frac{5x^2 - 40x + 45}{5x} = \frac{(x - 3)^2}{x}
$$

##### Step 3.2: Simplify $\frac{x^2 + 2x - 28}{x^2 - 3x + 2}$
- Factor the numerator $x^2 + 2x - 28$:
$$
x^2 + 2x - 28 = (x + 7)(x - 4)
$$
- Factor the denominator $x^2 - 3x + 2$:
$$
x^2 - 3x + 2 = (x - 1)(x - 2)
$$
- The fraction becomes:
$$
\frac{x^2 + 2x - 28}{x^2 - 3x + 2} = \frac{(x + 7)(x - 4)}{(x - 1)(x - 2)}
$$
- Since there are no common factors between the numerator and the denominator, the expression cannot be simplified further:
$$
\frac{x^2 + 2x - 28}{x^2 - 3x + 2} = \frac{(x + 7)(x - 4)}{(x - 1)(x - 2)}
$$

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#### 4. Simplify:
$$
\frac{5x^3 + 40x^2}{3x^4 + 18x^3} \quad \text{and} \quad \frac{3x^3 - 6x^2}{9x^2 - 18x}
$$

##### Step 4.1: Simplify $\frac{5x^3 + 40x^2}{3x^4 + 18x^3}$
- Factor the numerator $5x^3 + 40x^2$:
$$
5x^3 + 40x^2 = 5x^2(x + 8)
$$
- Factor the denominator $3x^4 + 18x^3$:
$$
3x^4 + 18x^3 = 3x^3(x + 6)
$$
- The fraction becomes:
$$
\frac{5x^3 + 40x^2}{3x^4 + 18x^3} = \frac{5x^2(x + 8)}{3x^3(x + 6)}
$$
- Cancel out the common factor $x^2$:
$$
\frac{5x^2(x + 8)}{3x^3(x + 6)} = \frac{5(x + 8)}{3x(x + 6)}
$$
- Therefore:
$$
\frac{5x^3 + 40x^2}{3x^4 + 18x^3} = \frac{5(x + 8)}{3x(x + 6)}
$$

##### Step 4.2: Simplify $\frac{3x^3 - 6x^2}{9x^2 - 18x}$
- Factor the numerator $3x^3 - 6x^2$:
$$
3x^3 - 6x^2 = 3x^2(x - 2)
$$
- Factor the denominator $9x^2 - 18x$:
$$
9x^2 - 18x = 9x(x - 2)
$$
- The fraction becomes:
$$
\frac{3x^3 - 6x^2}{9x^2 - 18x} = \frac{3x^2(x - 2)}{9x(x - 2)}
$$
- Cancel out the common factors $3x(x - 2)$:
$$
\frac{3x^2(x - 2)}{9x(x - 2)} = \frac{x}{3}
$$
- Therefore:
$$
\frac{3x^3 - 6x^2}{9x^2 - 18x} = \frac{x}{3}
$$

---

#### 5. Simplify:
$$
\frac{7x^2 + 7}{x^2 - 4x - 5} \quad \text{and} \quad \frac{9x^2 - 8x}{x^2 + 3x - 18}
$$

##### Step 5.1: Simplify $\frac{7x^2 + 7}{x^2 - 4x - 5}$
- Factor the numerator $7x^2 + 7$:
$$
7x^2 + 7 = 7(x^2 + 1)
$$
- Factor the denominator $x^2 - 4x - 5$:
$$
x^2 - 4x - 5 = (x - 5)(x + 1)
$$
- The fraction becomes:
$$
\frac{7x^2 + 7}{x^2 - 4x - 5} = \frac{7(x^2 + 1)}{(x - 5)(x + 1)}
$$
- Since there are no common factors between the numerator and the denominator, the expression cannot be simplified further:
$$
\frac{7x^2 + 7}{x^2 - 4x - 5} = \frac{7(x^2 + 1)}{(x - 5)(x + 1)}
$$

##### Step 5.2: Simplify $\frac{9x^2 - 8x}{x^2 + 3x - 18}$
- Factor the numerator $9x^2 - 8x$:
$$
9x^2 - 8x = x(9x - 8)
$$
- Factor the denominator $x^2 + 3x - 18$:
$$
x^2 + 3x - 18 = (x + 6)(x - 3)
$$
- The fraction becomes:
$$
\frac{9x^2 - 8x}{x^2 + 3x - 18} = \frac{x(9x - 8)}{(x + 6)(x - 3)}
$$
- Since there are no common factors between the numerator and the denominator, the expression cannot be simplified further:
$$
\frac{9x^2 - 8x}{x^2 + 3x - 18} = \frac{x(9x - 8)}{(x + 6)(x - 3)}
$$

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#### 6. Simplify:
$$
\frac{13x^2 + 13x}{8} \quad \text{and} \quad \frac{5x + 5}{x^2 - 12x + 35}
$$

##### Step 6.1: Simplify $\frac{13x^2 + 13x}{8}$
- Factor the numerator $13x^2 + 13x$:
$$
13x^2 + 13x = 13x(x + 1)
$$
- The fraction becomes:
$$
\frac{13x^2 + 13x}{8} = \frac{13x(x + 1)}{8}
$$
- Therefore:
$$
\frac{13x^2 + 13x}{8} = \frac{13x(x + 1)}{8}
$$

##### Step 6.2: Simplify $\frac{5x + 5}{x^2 - 12x + 35}$
- Factor the numerator $5x + 5$:
$$
5x + 5 = 5(x + 1)
$$
- Factor the denominator $x^2 - 12x + 35$:
$$
x^2 - 12x + 35 = (x - 7)(x - 5)
$$
- The fraction becomes:
$$
\frac{5x + 5}{x^2 - 12x + 35} = \frac{5(x + 1)}{(x - 7)(x - 5)}
$$
- Since there are no common factors between the numerator and the denominator, the expression cannot be simplified further:
$$
\frac{5x + 5}{x^2 - 12x + 35} = \frac{5(x + 1)}{(x - 7)(x - 5)}
$$

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#### 7. Simplify:
$$
\frac{x^2 - 13x + 42}{9x^2 - 54x} \quad \text{and} \quad \frac{18x^2 - 45x}{13x^2 - 45x}
$$

##### Step 7.1: Simplify $\frac{x^2 - 13x + 42}{9x^2 - 54x}$
- Factor the numerator $x^2 - 13x + 42$:
$$
x^2 - 13x + 42 = (x - 7)(x - 6)
$$
- Factor the denominator $9x^2 - 54x$:
$$
9x^2 - 54x = 9x(x - 6)
$$
- The fraction becomes:
$$
\frac{x^2 - 13x + 42}{9x^2 - 54x} = \frac{(x - 7)(x - 6)}{9x(x - 6)}
$$
- Cancel out the common factor $(x - 6)$:
$$
\frac{(x - 7)(x - 6)}{9x(x - 6)} = \frac{x - 7}{9x}
$$
- Therefore:
$$
\frac{x^2 - 13x + 42}{9x^2 - 54x} = \frac{x - 7}{9x}
$$

##### Step 7.2: Simplify $\frac{18x^2 - 45x}{13x^2 - 45x}$
- Factor the numerator $18x^2 - 45x$:
$$
18x^2 - 45x = 9x(2x - 5)
$$
- Factor the denominator $13x^2 - 45x$:
$$
13x^2 - 45x = x(13x - 45)
$$
- The fraction becomes:
$$
\frac{18x^2 - 45x}{13x^2 - 45x} = \frac{9x(2x - 5)}{x(13x - 45)}
$$
- Cancel out the common factor $x$:
$$
\frac{9x(2x - 5)}{x(13x - 45)} = \frac{9(2x - 5)}{13x - 45}
$$
- Therefore:
$$
\frac{18x^2 - 45x}{13x^2 - 45x} = \frac{9(2x - 5)}{13x - 45}
$$

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Final Answer:

$$
\boxed{
\begin{aligned}
1. & \quad \frac{x^2 - 18}{x + 3}, \quad \frac{3(5x - 18)}{x - 2} \\
2. & \quad \frac{(x - 8)(x + 2)}{3(x - 3)}, \quad \frac{x^2 + 4x + 5}{x^2 - 4x + 5} \\
3. & \quad \frac{(x - 3)^2}{x}, \quad \frac{(x + 7)(x - 4)}{(x - 1)(x - 2)} \\
4. & \quad \frac{5(x + 8)}{3x(x + 6)}, \quad \frac{x}{3} \\
5. & \quad \frac{7(x^2 + 1)}{(x - 5)(x + 1)}, \quad \frac{x(9x - 8)}{(x + 6)(x - 3)} \\
6. & \quad \frac{13x(x + 1)}{8}, \quad \frac{5(x + 1)}{(x - 7)(x - 5)} \\
7. & \quad \frac{x - 7}{9x}, \quad \frac{9(2x - 5)}{13x - 45}
\end{aligned}
}
$$