Finite Math Examples

$32 a^{2} - 8 a b - 12 b^{2}$

Step 1

Factor out the GCF of $4$ from $32 a^{2} - 8 a b - 12 b^{2}$ .

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Step 1.1

Factor out the GCF of $4$ from each term in the polynomial.

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Step 1.1.1

Factor out the GCF of $4$ from the expression $32 a^{2}$ .

$4 (8 a^{2}) - 8 a b - 12 b^{2}$

Step 1.1.2

Factor out the GCF of $4$ from the expression $- 8 a b$ .

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

Step 1.1.3

Factor out the GCF of $4$ from the expression $- 12 b^{2}$ .

$4 (8 a^{2}) + 4 (- 2 a b) + 4 (- 3 b^{2})$

Step 1.2

Since all the terms share a common factor of $4$ , it can be factored out of each term.

$4 (8 a^{2} - 2 a b - 3 b^{2})$

Step 2

Factor by grouping.

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Step 2.1

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 $a \cdot c = 8 \cdot - 3 = - 24$ and whose sum is $b = - 2$ .

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Step 2.1.1

Reorder terms.

$4 (8 a^{2} - 3 b^{2} - 2 a b)$

Step 2.1.2

Reorder $- 3 b^{2}$ and $- 2 a b$ .

$4 (8 a^{2} - 2 a b - 3 b^{2})$

Step 2.1.3

Factor $- 2$ out of $- 2 a b$ .

$4 (8 a^{2} - 2 (a b) - 3 b^{2})$

Step 2.1.4

Rewrite $- 2$ as $4$ plus $- 6$

$4 (8 a^{2} + (4 - 6) (a b) - 3 b^{2})$

Step 2.1.5

Apply the distributive property.

$4 (8 a^{2} + 4 (a b) - 6 (a b) - 3 b^{2})$

Step 2.1.6

Remove unnecessary parentheses.

$4 (8 a^{2} + 4 a b - 6 (a b) - 3 b^{2})$

Step 2.1.7

Remove unnecessary parentheses.

$4 (8 a^{2} + 4 a b - 6 a b - 3 b^{2})$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$4 ((8 a^{2} + 4 a b) - 6 a b - 3 b^{2})$

Step 2.2.2

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

$4 (4 a (2 a + b) - 3 b (2 a + b))$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $2 a + b$ .

$4 ((2 a + b) (4 a - 3 b))$