Divide ((5x^2-3x-14)/(3x^2+5x-2))÷((4x-8)/(3x-1))

$\frac{5 x^{2} - 3 x - 14}{3 x^{2} + 5 x - 2} \div \frac{4 x - 8}{3 x - 1}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{5 x^{2} - 3 x - 14}{3 x^{2} + 5 x - 2} \cdot \frac{3 x - 1}{4 x - 8}$

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 $a \cdot c = 5 \cdot - 14 = - 70$ and whose sum is $b = - 3$ .

Factor $- 3$ out of $- 3 x$ .

$\frac{5 x^{2} - 3 (x) - 14}{3 x^{2} + 5 x - 2} \cdot \frac{3 x - 1}{4 x - 8}$

Rewrite $- 3$ as $7$ plus $- 10$

$\frac{5 x^{2} + (7 - 10) x - 14}{3 x^{2} + 5 x - 2} \cdot \frac{3 x - 1}{4 x - 8}$

Apply the distributive property.

$\frac{5 x^{2} + 7 x - 10 x - 14}{3 x^{2} + 5 x - 2} \cdot \frac{3 x - 1}{4 x - 8}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(5 x^{2} + 7 x) - 10 x - 14}{3 x^{2} + 5 x - 2} \cdot \frac{3 x - 1}{4 x - 8}$

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

$\frac{x (5 x + 7) - 2 (5 x + 7)}{3 x^{2} + 5 x - 2} \cdot \frac{3 x - 1}{4 x - 8}$

Factor the polynomial by factoring out the greatest common factor, $5 x + 7$ .

$\frac{(5 x + 7) (x - 2)}{3 x^{2} + 5 x - 2} \cdot \frac{3 x - 1}{4 x - 8}$

Step 3

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

Factor $5$ out of $5 x$ .

$\frac{(5 x + 7) (x - 2)}{3 x^{2} + 5 (x) - 2} \cdot \frac{3 x - 1}{4 x - 8}$

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

$\frac{(5 x + 7) (x - 2)}{3 x^{2} + (- 1 + 6) x - 2} \cdot \frac{3 x - 1}{4 x - 8}$

Apply the distributive property.

$\frac{(5 x + 7) (x - 2)}{3 x^{2} - 1 x + 6 x - 2} \cdot \frac{3 x - 1}{4 x - 8}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(5 x + 7) (x - 2)}{(3 x^{2} - 1 x) + 6 x - 2} \cdot \frac{3 x - 1}{4 x - 8}$

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

$\frac{(5 x + 7) (x - 2)}{x (3 x - 1) + 2 (3 x - 1)} \cdot \frac{3 x - 1}{4 x - 8}$

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$\frac{(5 x + 7) (x - 2)}{(3 x - 1) (x + 2)} \cdot \frac{3 x - 1}{4 x - 8}$

Step 4

Cancel the common factor of $3 x - 1$ .

Cancel the common factor.

$\frac{(5 x + 7) (x - 2)}{(3 x - 1) (x + 2)} \cdot \frac{3 x - 1}{4 x - 8}$

Rewrite the expression.

$\frac{(5 x + 7) (x - 2)}{x + 2} \cdot \frac{1}{4 x - 8}$

Step 5

Multiply $\frac{(5 x + 7) (x - 2)}{x + 2}$ by $\frac{1}{4 x - 8}$ .

$\frac{(5 x + 7) (x - 2)}{(x + 2) (4 x - 8)}$

Step 6

Factor $4$ out of $4 x - 8$ .

Factor $4$ out of $4 x$ .

$\frac{(5 x + 7) (x - 2)}{(x + 2) (4 (x) - 8)}$

Factor $4$ out of $- 8$ .

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

Factor $4$ out of $4 x + 4 \cdot - 2$ .

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

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

Step 7

Cancel the common factor of $x - 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 8

Move $4$ to the left of $x + 2$ .

$\frac{5 x + 7}{4 (x + 2)}$

Divide ((5x^2-3x-14)/(3x^2+5x-2))÷((4x-8)/(3x-1))

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