Factor x^2-x^4+12

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

Step 1

Factor $- 1$ out of $x^{2} - x^{4} + 12$ .

Reorder $x^{2}$ and $- x^{4}$ .

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

Factor $- 1$ out of $- x^{4}$ .

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

Factor $- 1$ out of $x^{2}$ .

$- (x^{4}) - 1 (- x^{2}) + 12$

Rewrite $12$ as $- 1 (- 12)$ .

$- (x^{4}) - 1 (- x^{2}) - 1 (- 12)$

Factor $- 1$ out of $- (x^{4}) - 1 (- x^{2})$ .

$- (x^{4} - x^{2}) - 1 (- 12)$

Factor $- 1$ out of $- (x^{4} - x^{2}) - 1 (- 12)$ .

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

Step 2

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

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

Step 3

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

$- (u^{2} - u - 12)$

Step 4

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 $- 1$ .

$- 4, 3$

Write the factored form using these integers.

$- ((u - 4) (u + 3))$

Step 5

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

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

Step 6

Rewrite $4$ as $2^{2}$ .

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

Step 7

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

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

Remove unnecessary parentheses.

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

Factor x^2-x^4+12

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