Factor -9-x^4+10x^2

$- 9 - x^{4} + 10 x^{2}$

Step 1

Factor $- 1$ out of $- 9 - x^{4} + 10 x^{2}$ .

Move $- 9$ .

$- x^{4} + 10 x^{2} - 9$

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

$- (x^{4}) + 10 x^{2} - 9$

Factor $- 1$ out of $10 x^{2}$ .

$- (x^{4}) - (- 10 x^{2}) - 9$

Rewrite $- 9$ as $- 1 (9)$ .

$- (x^{4}) - (- 10 x^{2}) - 1 (9)$

Factor $- 1$ out of $- (x^{4}) - (- 10 x^{2})$ .

$- (x^{4} - 10 x^{2}) - 1 (9)$

Factor $- 1$ out of $- (x^{4} - 10 x^{2}) - 1 (9)$ .

$- (x^{4} - 10 x^{2} + 9)$

Step 2

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

$- ({(x^{2})}^{2} - 10 x^{2} + 9)$

Step 3

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

$- (u^{2} - 10 u + 9)$

Step 4

Factor $u^{2} - 10 u + 9$ 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 $9$ and whose sum is $- 10$ .

$- 9, - 1$

Write the factored form using these integers.

$- ((u - 9) (u - 1))$

Step 5

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

$- ((x^{2} - 9) (x^{2} - 1))$

Step 6

Rewrite $9$ as $3^{2}$ .

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

Step 7

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 $b = 3$ .

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

Step 8

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

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

Step 9

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 $b = 1$ .

$- ((x + 3) (x - 3) ((x + 1) (x - 1)))$

Remove unnecessary parentheses.

$- ((x + 3) (x - 3) (x + 1) (x - 1))$

Remove unnecessary parentheses.

$- (x + 3) (x - 3) (x + 1) (x - 1)$

Factor -9-x^4+10x^2

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