Factor x^4+2x^3+x^2+8x-12

$x^{4} + 2 x^{3} + x^{2} + 8 x - 12$

Regroup terms.

$2 x^{3} + 8 x + x^{4} + x^{2} - 12$

Factor $2 x$ out of $2 x^{3} + 8 x$ .

Factor $2 x$ out of $2 x^{3}$ .

$2 x (x^{2}) + 8 x + x^{4} + x^{2} - 12$

Factor $2 x$ out of $8 x$ .

$2 x (x^{2}) + 2 x (4) + x^{4} + x^{2} - 12$

Factor $2 x$ out of $2 x (x^{2}) + 2 x (4)$ .

$2 x (x^{2} + 4) + x^{4} + x^{2} - 12$

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$2 x (x^{2} + 4) + {(x^{2})}^{2} + x^{2} - 12$

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

$2 x (x^{2} + 4) + u^{2} + u - 12$

Factor $u^{2} + u - 12$ 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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Write the factored form using these integers.

$2 x (x^{2} + 4) + (u - 3) (u + 4)$

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

$2 x (x^{2} + 4) + (x^{2} - 3) (x^{2} + 4)$

Factor $x^{2} + 4$ out of $2 x (x^{2} + 4) + (x^{2} - 3) (x^{2} + 4)$ .

Factor $x^{2} + 4$ out of $2 x (x^{2} + 4)$ .

$(x^{2} + 4) (2 x) + (x^{2} - 3) (x^{2} + 4)$

Factor $x^{2} + 4$ out of $(x^{2} - 3) (x^{2} + 4)$ .

$(x^{2} + 4) (2 x) + (x^{2} + 4) (x^{2} - 3)$

Factor $x^{2} + 4$ out of $(x^{2} + 4) (2 x) + (x^{2} + 4) (x^{2} - 3)$ .

$(x^{2} + 4) (2 x + x^{2} - 3)$

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$(x^{2} + 4) (2 u + u^{2} - 3)$

Factor $2 u + u^{2} - 3$ 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.

$(x^{2} + 4) ((u - 1) (u + 3))$

Factor.

Replace all occurrences of $u$ with $x$ .

$(x^{2} + 4) ((x - 1) (x + 3))$

Remove unnecessary parentheses.

$(x^{2} + 4) (x - 1) (x + 3)$

Factor x^4+2x^3+x^2+8x-12

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