Factor (a^2+2a)^2-2(a^2+2a)-3

${(a^{2} + 2 a)}^{2} - 2 (a^{2} + 2 a) - 3$

Step 1

Let $u = a^{2} + 2 a$ . Substitute $u$ for all occurrences of $a^{2} + 2 a$ .

$u^{2} - 2 u - 3$

Step 2

Factor $u^{2} - 2 u - 3$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Write the factored form using these integers.

$(u - 3) (u + 1)$

Step 3

Replace all occurrences of $u$ with $a^{2} + 2 a$ .

$(a^{2} + 2 a - 3) (a^{2} + 2 a + 1)$

Step 4

Factor $a^{2} + 2 a - 3$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $2$ .

$- 1, 3$

Write the factored form using these integers.

$(a - 1) (a + 3) (a^{2} + 2 a + 1)$

Step 5

Factor using the perfect square rule.

Rewrite $1$ as $1^{2}$ .

$(a - 1) (a + 3) (a^{2} + 2 a + 1^{2})$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a = 2 \cdot a \cdot 1$

Rewrite the polynomial.

$(a - 1) (a + 3) (a^{2} + 2 \cdot a \cdot 1 + 1^{2})$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = a$ and $b = 1$ .

$(a - 1) (a + 3) {(a + 1)}^{2}$

Factor (a^2+2a)^2-2(a^2+2a)-3

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