Simplify (4x^2-9)/(3x^2-19x-14)*(2x-14)/(2x-3)

$\frac{4 x^{2} - 9}{3 x^{2} - 19 x - 14} \cdot \frac{2 x - 14}{2 x - 3}$

Step 1

Simplify the numerator.

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$\frac{{(2 x)}^{2} - 9}{3 x^{2} - 19 x - 14} \cdot \frac{2 x - 14}{2 x - 3}$

Rewrite $9$ as $3^{2}$ .

$\frac{{(2 x)}^{2} - 3^{2}}{3 x^{2} - 19 x - 14} \cdot \frac{2 x - 14}{2 x - 3}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 3$ .

$\frac{(2 x + 3) (2 x - 3)}{3 x^{2} - 19 x - 14} \cdot \frac{2 x - 14}{2 x - 3}$

Step 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 $a \cdot c = 3 \cdot - 14 = - 42$ and whose sum is $b = - 19$ .

Factor $- 19$ out of $- 19 x$ .

$\frac{(2 x + 3) (2 x - 3)}{3 x^{2} - 19 (x) - 14} \cdot \frac{2 x - 14}{2 x - 3}$

Rewrite $- 19$ as $2$ plus $- 21$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

Step 3

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

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $- 14$ .

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

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

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

Step 4

Cancel the common factor of $2 x - 3$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Cancel the common factor of $x - 7$ .

Factor $x - 7$ out of $(3 x + 2) (x - 7)$ .

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

Factor $x - 7$ out of $2 (x - 7)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{2 x + 3}{3 x + 2} \cdot 2$

Step 6

Combine $\frac{2 x + 3}{3 x + 2}$ and $2$ .

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

Step 7

Move $2$ to the left of $2 x + 3$ .

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

Simplify (4x^2-9)/(3x^2-19x-14)*(2x-14)/(2x-3)

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