Simplify ((25x^2-4)/(2x^2-11x-21))÷((5x-2)/(2x-14))

$\frac{25 x^{2} - 4}{2 x^{2} - 11 x - 21} \div \frac{5 x - 2}{2 x - 14}$

To divide by a fraction, multiply by its reciprocal.

$\frac{25 x^{2} - 4}{2 x^{2} - 11 x - 21} \cdot \frac{2 x - 14}{5 x - 2}$

Simplify the numerator.

Rewrite $25 x^{2}$ as ${(5 x)}^{2}$ .

$\frac{{(5 x)}^{2} - 4}{2 x^{2} - 11 x - 21} \cdot \frac{2 x - 14}{5 x - 2}$

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

$\frac{{(5 x)}^{2} - 2^{2}}{2 x^{2} - 11 x - 21} \cdot \frac{2 x - 14}{5 x - 2}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5 x$ and $b = 2$ .

$\frac{(5 x + 2) (5 x - 2)}{2 x^{2} - 11 x - 21} \cdot \frac{2 x - 14}{5 x - 2}$

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 $a \cdot c = 2 \cdot - 21 = - 42$ and whose sum is $b = - 11$ .

Factor $- 11$ out of $- 11 x$ .

$\frac{(5 x + 2) (5 x - 2)}{2 x^{2} - 11 (x) - 21} \cdot \frac{2 x - 14}{5 x - 2}$

Rewrite $- 11$ as $3$ plus $- 14$

$\frac{(5 x + 2) (5 x - 2)}{2 x^{2} + (3 - 14) x - 21} \cdot \frac{2 x - 14}{5 x - 2}$

Apply the distributive property.

$\frac{(5 x + 2) (5 x - 2)}{2 x^{2} + 3 x - 14 x - 21} \cdot \frac{2 x - 14}{5 x - 2}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(5 x + 2) (5 x - 2)}{(2 x^{2} + 3 x) - 14 x - 21} \cdot \frac{2 x - 14}{5 x - 2}$

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

$\frac{(5 x + 2) (5 x - 2)}{x (2 x + 3) - 7 (2 x + 3)} \cdot \frac{2 x - 14}{5 x - 2}$

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

$\frac{(5 x + 2) (5 x - 2)}{(2 x + 3) (x - 7)} \cdot \frac{2 x - 14}{5 x - 2}$

Cancel the common factor of $5 x - 2$ .

Factor $5 x - 2$ out of $(5 x + 2) (5 x - 2)$ .

$\frac{(5 x - 2) (5 x + 2)}{(2 x + 3) (x - 7)} \cdot \frac{2 x - 14}{5 x - 2}$

Cancel the common factor.

$\frac{(5 x - 2) (5 x + 2)}{(2 x + 3) (x - 7)} \cdot \frac{2 x - 14}{5 x - 2}$

Rewrite the expression.

$\frac{5 x + 2}{(2 x + 3) (x - 7)} (2 x - 14)$

Multiply $\frac{5 x + 2}{(2 x + 3) (x - 7)}$ by $2 x - 14$ .

$\frac{(5 x + 2) (2 x - 14)}{(2 x + 3) (x - 7)}$

Factor $2$ out of $2 x - 14$ .

Factor $2$ out of $2 x$ .

$\frac{(5 x + 2) (2 (x) - 14)}{(2 x + 3) (x - 7)}$

Factor $2$ out of $- 14$ .

$\frac{(5 x + 2) (2 x + 2 \cdot - 7)}{(2 x + 3) (x - 7)}$

Factor $2$ out of $2 x + 2 \cdot - 7$ .

$\frac{(5 x + 2) (2 (x - 7))}{(2 x + 3) (x - 7)}$

$\frac{(5 x + 2) \cdot 2 (x - 7)}{(2 x + 3) (x - 7)}$

Cancel the common factor of $x - 7$ .

Cancel the common factor.

$\frac{(5 x + 2) \cdot 2 (x - 7)}{(2 x + 3) (x - 7)}$

Rewrite the expression.

$\frac{(5 x + 2) \cdot 2}{2 x + 3}$

Move $2$ to the left of $5 x + 2$ .

$\frac{2 (5 x + 2)}{2 x + 3}$

Simplify ((25x^2-4)/(2x^2-11x-21))÷((5x-2)/(2x-14))

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