Factor (4x^2+5x)^2-5(4x^2+5x)-6

${(4 x^{2} + 5 x)}^{2} - 5 (4 x^{2} + 5 x) - 6$

Step 1

Let $u = 4 x^{2} + 5 x$ . Substitute $u$ for all occurrences of $4 x^{2} + 5 x$ .

$u^{2} - 5 u - 6$

Step 2

Factor $u^{2} - 5 u - 6$ 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 $- 6$ and whose sum is $- 5$ .

$- 6, 1$

Write the factored form using these integers.

$(u - 6) (u + 1)$

Step 3

Replace all occurrences of $u$ with $4 x^{2} + 5 x$ .

$(4 x^{2} + 5 x - 6) (4 x^{2} + 5 x + 1)$

Step 4

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot - 6 = - 24$ and whose sum is $b = 5$ .

Factor $5$ out of $5 x$ .

$(4 x^{2} + 5 (x) - 6) (4 x^{2} + 5 x + 1)$

Rewrite $5$ as $- 3$ plus $8$

$(4 x^{2} + (- 3 + 8) x - 6) (4 x^{2} + 5 x + 1)$

Apply the distributive property.

$(4 x^{2} - 3 x + 8 x - 6) (4 x^{2} + 5 x + 1)$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$((4 x^{2} - 3 x) + 8 x - 6) (4 x^{2} + 5 x + 1)$

Factor out the greatest common factor (GCF) from each group.

$(x (4 x - 3) + 2 (4 x - 3)) (4 x^{2} + 5 x + 1)$

Factor the polynomial by factoring out the greatest common factor, $4 x - 3$ .

$(4 x - 3) (x + 2) (4 x^{2} + 5 x + 1)$

Step 5

Factor.

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot 1 = 4$ and whose sum is $b = 5$ .

Factor $5$ out of $5 x$ .

$(4 x - 3) (x + 2) (4 x^{2} + 5 (x) + 1)$

Rewrite $5$ as $1$ plus $4$

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

Apply the distributive property.

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

Multiply $x$ by $1$ .

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

Factor out the greatest common factor (GCF) from each group.

$(4 x - 3) (x + 2) (x (4 x + 1) + 1 (4 x + 1))$

Factor the polynomial by factoring out the greatest common factor, $4 x + 1$ .

$(4 x - 3) (x + 2) ((4 x + 1) (x + 1))$

Remove unnecessary parentheses.

$(4 x - 3) (x + 2) (4 x + 1) (x + 1)$

Factor (4x^2+5x)^2-5(4x^2+5x)-6

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