Simplify (x^2+2x-3)/((x-4)/((2x^2+5x-3)/(x^2-16)))

$\frac{x^{2} + 2 x - 3}{\frac{x - 4}{\frac{2 x^{2} + 5 x - 3}{x^{2} - 16}}}$

Multiply the numerator by the reciprocal of the denominator.

$(x^{2} + 2 x - 3) \frac{\frac{2 x^{2} + 5 x - 3}{x^{2} - 16}}{x - 4}$

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 $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = 5$ .

Factor $5$ out of $5 x$ .

$(x^{2} + 2 x - 3) \frac{\frac{2 x^{2} + 5 (x) - 3}{x^{2} - 16}}{x - 4}$

Rewrite $5$ as $- 1$ plus $6$

$(x^{2} + 2 x - 3) \frac{\frac{2 x^{2} + (- 1 + 6) x - 3}{x^{2} - 16}}{x - 4}$

Apply the distributive property.

$(x^{2} + 2 x - 3) \frac{\frac{2 x^{2} - 1 x + 6 x - 3}{x^{2} - 16}}{x - 4}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x^{2} + 2 x - 3) \frac{\frac{(2 x^{2} - 1 x) + 6 x - 3}{x^{2} - 16}}{x - 4}$

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

$(x^{2} + 2 x - 3) \frac{\frac{x (2 x - 1) + 3 (2 x - 1)}{x^{2} - 16}}{x - 4}$

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$(x^{2} + 2 x - 3) \frac{\frac{(2 x - 1) (x + 3)}{x^{2} - 16}}{x - 4}$

Simplify the denominator.

Rewrite $16$ as $4^{2}$ .

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

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 $b = 4$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Simplify the numerator.

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 $2$ .

$- 1, 3$

Write the factored form using these integers.

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

Combine exponents.

Raise $x + 3$ to the power of $1$ .

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

Raise $x + 3$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Simplify (x^2+2x-3)/((x-4)/((2x^2+5x-3)/(x^2-16)))

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