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

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

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.

$\frac{(x - 1) (x + 3)}{x - 4} \cdot \frac{x^{2} - 16}{2 x^{2} + 5 x - 3}$

Step 3

Simplify the numerator.

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

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

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 $b = 4$ .

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

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

Factor $5$ out of $5 x$ .

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

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

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

Step 5

Cancel the common factor of $x + 3$ .

Factor $x + 3$ out of $(x - 1) (x + 3)$ .

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

Factor $x + 3$ out of $(2 x - 1) (x + 3)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

Cancel the common factor of $x - 4$ .

Factor $x - 4$ out of $(x + 4) (x - 4)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 7

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

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

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

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