Factor x^5+x^4-3x^3-3x^2-4x-4

$x^{5} + x^{4} - 3 x^{3} - 3 x^{2} - 4 x - 4$

Step 1

Regroup terms.

$x^{5} - 3 x^{3} - 4 x + x^{4} - 3 x^{2} - 4$

Step 2

Factor $x$ out of $x^{5} - 3 x^{3} - 4 x$ .

Factor $x$ out of $x^{5}$ .

$x \cdot x^{4} - 3 x^{3} - 4 x + x^{4} - 3 x^{2} - 4$

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

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

Factor $x$ out of $- 4 x$ .

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

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

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

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

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

Step 3

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

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

Step 4

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

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

Step 5

Factor $u^{2} - 3 u - 4$ 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 $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Write the factored form using these integers.

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

Step 6

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

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

Step 7

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

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

Step 8

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 $b = 2$ .

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

Remove unnecessary parentheses.

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

Step 9

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

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

Step 10

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

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

Step 11

Factor $u^{2} - 3 u - 4$ 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 $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Write the factored form using these integers.

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

Step 12

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

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

Step 13

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

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

Step 14

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 $b = 2$ .

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

Step 15

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

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

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

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

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

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

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

Factor x^5+x^4-3x^3-3x^2-4x-4

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