Simplify ((2x^2+3x-14)/(3x^2-2x-5))÷((2x-4)/(3x-5))

$\frac{2 x^{2} + 3 x - 14}{3 x^{2} - 2 x - 5} \div \frac{2 x - 4}{3 x - 5}$

To divide by a fraction, multiply by its reciprocal.

$\frac{2 x^{2} + 3 x - 14}{3 x^{2} - 2 x - 5} \cdot \frac{3 x - 5}{2 x - 4}$

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 $a \cdot c = 2 \cdot - 14 = - 28$ and whose sum is $b = 3$ .

Factor $3$ out of $3 x$ .

$\frac{2 x^{2} + 3 (x) - 14}{3 x^{2} - 2 x - 5} \cdot \frac{3 x - 5}{2 x - 4}$

Rewrite $3$ as $- 4$ plus $7$

$\frac{2 x^{2} + (- 4 + 7) x - 14}{3 x^{2} - 2 x - 5} \cdot \frac{3 x - 5}{2 x - 4}$

Apply the distributive property.

$\frac{2 x^{2} - 4 x + 7 x - 14}{3 x^{2} - 2 x - 5} \cdot \frac{3 x - 5}{2 x - 4}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(2 x^{2} - 4 x) + 7 x - 14}{3 x^{2} - 2 x - 5} \cdot \frac{3 x - 5}{2 x - 4}$

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

$\frac{2 x (x - 2) + 7 (x - 2)}{3 x^{2} - 2 x - 5} \cdot \frac{3 x - 5}{2 x - 4}$

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

$\frac{(x - 2) (2 x + 7)}{3 x^{2} - 2 x - 5} \cdot \frac{3 x - 5}{2 x - 4}$

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 $a \cdot c = 3 \cdot - 5 = - 15$ and whose sum is $b = - 2$ .

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

$\frac{(x - 2) (2 x + 7)}{3 x^{2} - 2 (x) - 5} \cdot \frac{3 x - 5}{2 x - 4}$

Rewrite $- 2$ as $3$ plus $- 5$

$\frac{(x - 2) (2 x + 7)}{3 x^{2} + (3 - 5) x - 5} \cdot \frac{3 x - 5}{2 x - 4}$

Apply the distributive property.

$\frac{(x - 2) (2 x + 7)}{3 x^{2} + 3 x - 5 x - 5} \cdot \frac{3 x - 5}{2 x - 4}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(x - 2) (2 x + 7)}{(3 x^{2} + 3 x) - 5 x - 5} \cdot \frac{3 x - 5}{2 x - 4}$

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

$\frac{(x - 2) (2 x + 7)}{3 x (x + 1) - 5 (x + 1)} \cdot \frac{3 x - 5}{2 x - 4}$

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

$\frac{(x - 2) (2 x + 7)}{(x + 1) (3 x - 5)} \cdot \frac{3 x - 5}{2 x - 4}$

Cancel the common factor of $3 x - 5$ .

Factor $3 x - 5$ out of $(x + 1) (3 x - 5)$ .

$\frac{(x - 2) (2 x + 7)}{(3 x - 5) (x + 1)} \cdot \frac{3 x - 5}{2 x - 4}$

Cancel the common factor.

$\frac{(x - 2) (2 x + 7)}{(3 x - 5) (x + 1)} \cdot \frac{3 x - 5}{2 x - 4}$

Rewrite the expression.

$\frac{(x - 2) (2 x + 7)}{x + 1} \cdot \frac{1}{2 x - 4}$

Multiply $\frac{(x - 2) (2 x + 7)}{x + 1}$ by $\frac{1}{2 x - 4}$ .

$\frac{(x - 2) (2 x + 7)}{(x + 1) (2 x - 4)}$

Factor $2$ out of $2 x - 4$ .

Factor $2$ out of $2 x$ .

$\frac{(x - 2) (2 x + 7)}{(x + 1) (2 (x) - 4)}$

Factor $2$ out of $- 4$ .

$\frac{(x - 2) (2 x + 7)}{(x + 1) (2 x + 2 \cdot - 2)}$

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

$\frac{(x - 2) (2 x + 7)}{(x + 1) (2 (x - 2))}$

$\frac{(x - 2) (2 x + 7)}{(x + 1) \cdot 2 (x - 2)}$

Cancel the common factor of $x - 2$ .

Cancel the common factor.

$\frac{(x - 2) (2 x + 7)}{(x + 1) \cdot 2 (x - 2)}$

Rewrite the expression.

$\frac{2 x + 7}{(x + 1) \cdot 2}$

Move $2$ to the left of $x + 1$ .

$\frac{2 x + 7}{2 (x + 1)}$

Simplify ((2x^2+3x-14)/(3x^2-2x-5))÷((2x-4)/(3x-5))

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