Simplify (x^2+x-12)/((x-2)/((3x^2+11x-4)/(x^2-4)))

$\frac{x^{2} + x - 12}{\frac{x - 2}{\frac{3 x^{2} + 11 x - 4}{x^{2} - 4}}}$

Multiply the numerator by the reciprocal of the denominator.

$(x^{2} + x - 12) \frac{\frac{3 x^{2} + 11 x - 4}{x^{2} - 4}}{x - 2}$

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 $a \cdot c = 3 \cdot - 4 = - 12$ and whose sum is $b = 11$ .

Factor $11$ out of $11 x$ .

$(x^{2} + x - 12) \frac{\frac{3 x^{2} + 11 (x) - 4}{x^{2} - 4}}{x - 2}$

Rewrite $11$ as $- 1$ plus $12$

$(x^{2} + x - 12) \frac{\frac{3 x^{2} + (- 1 + 12) x - 4}{x^{2} - 4}}{x - 2}$

Apply the distributive property.

$(x^{2} + x - 12) \frac{\frac{3 x^{2} - 1 x + 12 x - 4}{x^{2} - 4}}{x - 2}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x^{2} + x - 12) \frac{\frac{(3 x^{2} - 1 x) + 12 x - 4}{x^{2} - 4}}{x - 2}$

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

$(x^{2} + x - 12) \frac{\frac{x (3 x - 1) + 4 (3 x - 1)}{x^{2} - 4}}{x - 2}$

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$(x^{2} + x - 12) \frac{\frac{(3 x - 1) (x + 4)}{x^{2} - 4}}{x - 2}$

Simplify the denominator.

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

$(x^{2} + x - 12) \frac{\frac{(3 x - 1) (x + 4)}{x^{2} - 2^{2}}}{x - 2}$

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 $b = 2$ .

$(x^{2} + x - 12) \frac{\frac{(3 x - 1) (x + 4)}{(x + 2) (x - 2)}}{x - 2}$

Multiply the numerator by the reciprocal of the denominator.

$(x^{2} + x - 12) (\frac{(3 x - 1) (x + 4)}{(x + 2) (x - 2)} \cdot \frac{1}{x - 2})$

Multiply $\frac{(3 x - 1) (x + 4)}{(x + 2) (x - 2)} \cdot \frac{1}{x - 2}$ .

Multiply $\frac{(3 x - 1) (x + 4)}{(x + 2) (x - 2)}$ by $\frac{1}{x - 2}$ .

$(x^{2} + x - 12) \frac{(3 x - 1) (x + 4)}{(x + 2) (x - 2) (x - 2)}$

Raise $x - 2$ to the power of $1$ .

$(x^{2} + x - 12) \frac{(3 x - 1) (x + 4)}{(x + 2) ({(x - 2)}^{1} (x - 2))}$

Raise $x - 2$ to the power of $1$ .

$(x^{2} + x - 12) \frac{(3 x - 1) (x + 4)}{(x + 2) ({(x - 2)}^{1} {(x - 2)}^{1})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{2} + x - 12) \frac{(3 x - 1) (x + 4)}{(x + 2) {(x - 2)}^{1 + 1}}$

Add $1$ and $1$ .

$(x^{2} + x - 12) \frac{(3 x - 1) (x + 4)}{(x + 2) {(x - 2)}^{2}}$

Multiply $x^{2} + x - 12$ by $\frac{(3 x - 1) (x + 4)}{(x + 2) {(x - 2)}^{2}}$ .

$\frac{(x^{2} + x - 12) ((3 x - 1) (x + 4))}{(x + 2) {(x - 2)}^{2}}$

Simplify the numerator.

Factor $x^{2} + x - 12$ 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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Write the factored form using these integers.

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

Combine exponents.

Raise $x + 4$ to the power of $1$ .

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

Raise $x + 4$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

Simplify (x^2+x-12)/((x-2)/((3x^2+11x-4)/(x^2-4)))

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