Simplify (4x^2+2x-12)/(4x^2-8x-12)

$\frac{4 x^{2} + 2 x - 12}{4 x^{2} - 8 x - 12}$

Cancel the common factor of $4 x^{2} + 2 x - 12$ and $4 x^{2} - 8 x - 12$ .

Factor $2$ out of $4 x^{2}$ .

$\frac{2 (2 x^{2}) + 2 x - 12}{4 x^{2} - 8 x - 12}$

Factor $2$ out of $2 x$ .

$\frac{2 (2 x^{2}) + 2 (x) - 12}{4 x^{2} - 8 x - 12}$

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

$\frac{2 (2 x^{2} + x) - 12}{4 x^{2} - 8 x - 12}$

Factor $2$ out of $- 12$ .

$\frac{2 (2 x^{2} + x) + 2 (- 6)}{4 x^{2} - 8 x - 12}$

Factor $2$ out of $2 (2 x^{2} + x) + 2 (- 6)$ .

$\frac{2 (2 x^{2} + x - 6)}{4 x^{2} - 8 x - 12}$

Cancel the common factors.

Factor $2$ out of $4 x^{2}$ .

$\frac{2 (2 x^{2} + x - 6)}{2 (2 x^{2}) - 8 x - 12}$

Factor $2$ out of $- 8 x$ .

$\frac{2 (2 x^{2} + x - 6)}{2 (2 x^{2}) + 2 (- 4 x) - 12}$

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

$\frac{2 (2 x^{2} + x - 6)}{2 (2 x^{2} - 4 x) - 12}$

Factor $2$ out of $- 12$ .

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

Factor $2$ out of $2 (2 x^{2} - 4 x) + 2 (- 6)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{2 x^{2} + x - 6}{2 x^{2} - 4 x - 6}$

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 $a \cdot c = 2 \cdot - 6 = - 12$ and whose sum is $b = 1$ .

Multiply by $1$ .

$\frac{2 x^{2} + 1 x - 6}{2 x^{2} - 4 x - 6}$

Rewrite $1$ as $- 3$ plus $4$

$\frac{2 x^{2} + (- 3 + 4) x - 6}{2 x^{2} - 4 x - 6}$

Apply the distributive property.

$\frac{2 x^{2} - 3 x + 4 x - 6}{2 x^{2} - 4 x - 6}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(2 x^{2} - 3 x) + 4 x - 6}{2 x^{2} - 4 x - 6}$

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

$\frac{x (2 x - 3) + 2 (2 x - 3)}{2 x^{2} - 4 x - 6}$

Factor the polynomial by factoring out the greatest common factor, $2 x - 3$ .

$\frac{(2 x - 3) (x + 2)}{2 x^{2} - 4 x - 6}$

Simplify the denominator.

Factor $2$ out of $2 x^{2} - 4 x - 6$ .

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

$\frac{(2 x - 3) (x + 2)}{2 (x^{2}) - 4 x - 6}$

Factor $2$ out of $- 4 x$ .

$\frac{(2 x - 3) (x + 2)}{2 (x^{2}) + 2 (- 2 x) - 6}$

Factor $2$ out of $- 6$ .

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

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

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

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

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

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

$- 3, 1$

Write the factored form using these integers.

$\frac{(2 x - 3) (x + 2)}{2 ((x - 3) (x + 1))}$

$\frac{(2 x - 3) (x + 2)}{2 (x - 3) (x + 1)}$

Simplify (4x^2+2x-12)/(4x^2-8x-12)

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