List of algebraic expressions for factoring practice.

It looks like you've uploaded a list of algebraic expressions, and the task is likely to factor each expression. Let's go through each one step by step and factor them completely where possible.

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1. $ x^2 + 2xy + y^2 $

This is a perfect square trinomial:
$$
x^2 + 2xy + y^2 = (x + y)^2
$$

✔ Answer: $ (x + y)^2 $

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2. $ a^2 + 2a + 1 $

Another perfect square trinomial:
$$
a^2 + 2a + 1 = (a + 1)^2
$$

✔ Answer: $ (a + 1)^2 $

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3. $ 2x^2 - 98 $

Factor out the GCF first:
$$
2x^2 - 98 = 2(x^2 - 49)
$$
Now, $ x^2 - 49 $ is a difference of squares:
$$
x^2 - 49 = (x - 7)(x + 7)
$$
So:
$$
2x^2 - 98 = 2(x - 7)(x + 7)
$$

✔ Answer: $ 2(x - 7)(x + 7) $

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4. $ x^2 + 3x + xy + 3y $

Group terms:
$$
(x^2 + 3x) + (xy + 3y)
$$
Factor each group:
$$
x(x + 3) + y(x + 3)
$$
Now factor out the common binomial:
$$
(x + y)(x + 3)
$$

✔ Answer: $ (x + y)(x + 3) $

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5. $ 20 - 3x - 2x^2 $

Rewrite in standard form:
$$
-2x^2 - 3x + 20
$$
Factor out $-1$ to make leading coefficient positive:
$$
-(2x^2 + 3x - 20)
$$
Now factor $2x^2 + 3x - 20$:
We look for two numbers that multiply to $2 \times (-20) = -40$, add to $3$:
These are $8$ and $-5$

Split the middle term:
$$
2x^2 + 8x - 5x - 20 = 2x(x + 4) -5(x + 4) = (2x - 5)(x + 4)
$$
So:
$$
-2x^2 - 3x + 20 = -(2x - 5)(x + 4)
$$

✔ Answer: $ -(2x - 5)(x + 4) $

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6. $ 4x^2 - x - 5 $

Use factoring by grouping or AC method:

AC = $4 \times (-5) = -20$
Find two numbers that multiply to $-20$, add to $-1$: $-5$ and $4$

Split middle term:
$$
4x^2 - 5x + 4x - 5 = x(4x - 5) + 1(4x - 5) = (x + 1)(4x - 5)
$$

✔ Answer: $ (x + 1)(4x - 5) $

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7. $ 9 - y^2 $

Difference of squares:
$$
9 - y^2 = (3)^2 - (y)^2 = (3 - y)(3 + y)
$$

✔ Answer: $ (3 - y)(3 + y) $ or $ (3 + y)(3 - y) $

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8. $ 7x^2 + 19x - 6 $

AC = $7 \times (-6) = -42$
Find two numbers that multiply to $-42$, add to $19$: $21$ and $-2$

Split:
$$
7x^2 + 21x - 2x - 6 = 7x(x + 3) -2(x + 3) = (7x - 2)(x + 3)
$$

✔ Answer: $ (7x - 2)(x + 3) $

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9. $ y^2 + 6y - 16 $

Find two numbers that multiply to $-16$, add to $6$: $8$ and $-2$

$$
y^2 + 8y - 2y - 16 = y(y + 8) -2(y + 8) = (y - 2)(y + 8)
$$

✔ Answer: $ (y - 2)(y + 8) $

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10. $ 14t^2 - 9t + 1 $

AC = $14 \times 1 = 14$
Find two numbers that multiply to $14$, add to $-9$: $-7$ and $-2$

Split:
$$
14t^2 - 7t - 2t + 1 = 7t(2t - 1) -1(2t - 1) = (7t - 1)(2t - 1)
$$

✔ Answer: $ (7t - 1)(2t - 1) $

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11. $ 6x^3 - 6x^2 - 120x $

Factor out GCF: $6x$
$$
6x(x^2 - x - 20)
$$
Now factor quadratic: find two numbers that multiply to $-20$, add to $-1$: $-5$ and $4$

$$
x^2 - x - 20 = (x - 5)(x + 4)
$$
So:
$$
6x(x - 5)(x + 4)
$$

✔ Answer: $ 6x(x - 5)(x + 4) $

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12. $ 125 - 8y^3 $

Recognize as a difference of cubes:
$$
125 = 5^3,\quad 8y^3 = (2y)^3
$$
Use identity: $ a^3 - b^3 = (a - b)(a^2 + ab + b^2) $

So:
$$
125 - 8y^3 = (5)^3 - (2y)^3 = (5 - 2y)(25 + 10y + 4y^2)
$$

✔ Answer: $ (5 - 2y)(25 + 10y + 4y^2) $

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13. $ x^2 + 14x - 32 $

Find two numbers that multiply to $-32$, add to $14$: $16$ and $-2$

$$
x^2 + 16x - 2x - 32 = x(x + 16) -2(x + 16) = (x - 2)(x + 16)
$$

✔ Answer: $ (x - 2)(x + 16) $

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✔ Final Answers Summary:

1. $ (x + y)^2 $
2. $ (a + 1)^2 $
3. $ 2(x - 7)(x + 7) $
4. $ (x + y)(x + 3) $
5. $ -(2x - 5)(x + 4) $
6. $ (x + 1)(4x - 5) $
7. $ (3 - y)(3 + y) $
8. $ (7x - 2)(x + 3) $
9. $ (y - 2)(y + 8) $
10. $ (7t - 1)(2t - 1) $
11. $ 6x(x - 5)(x + 4) $
12. $ (5 - 2y)(25 + 10y + 4y^2) $
13. $ (x - 2)(x + 16) $

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