Simplify ((-x^2+x+20)/(5x^2-25x))÷((x+4)/(2x-14))

$\frac{- x^{2} + x + 20}{5 x^{2} - 25 x} \div \frac{x + 4}{2 x - 14}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{- x^{2} + x + 20}{5 x^{2} - 25 x} \cdot \frac{2 x - 14}{x + 4}$

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 $a \cdot c = - 1 \cdot 20 = - 20$ and whose sum is $b = 1$ .

Multiply by $1$ .

$\frac{- x^{2} + 1 x + 20}{5 x^{2} - 25 x} \cdot \frac{2 x - 14}{x + 4}$

Rewrite $1$ as $- 4$ plus $5$

$\frac{- x^{2} + (- 4 + 5) x + 20}{5 x^{2} - 25 x} \cdot \frac{2 x - 14}{x + 4}$

Apply the distributive property.

$\frac{- x^{2} - 4 x + 5 x + 20}{5 x^{2} - 25 x} \cdot \frac{2 x - 14}{x + 4}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(- x^{2} - 4 x) + 5 x + 20}{5 x^{2} - 25 x} \cdot \frac{2 x - 14}{x + 4}$

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

$\frac{x (- x - 4) - 5 (- x - 4)}{5 x^{2} - 25 x} \cdot \frac{2 x - 14}{x + 4}$

Factor the polynomial by factoring out the greatest common factor, $- x - 4$ .

$\frac{(- x - 4) (x - 5)}{5 x^{2} - 25 x} \cdot \frac{2 x - 14}{x + 4}$

Step 3

Factor $5 x$ out of $5 x^{2} - 25 x$ .

Factor $5 x$ out of $5 x^{2}$ .

$\frac{(- x - 4) (x - 5)}{5 x (x) - 25 x} \cdot \frac{2 x - 14}{x + 4}$

Factor $5 x$ out of $- 25 x$ .

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

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

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

Step 4

Cancel the common factor of $x - 5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

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

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $- 14$ .

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

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

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

Step 6

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

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

Step 7

Cancel the common factor of $- x - 4$ and $x + 4$ .

Factor $- 1$ out of $- x$ .

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

Rewrite $- 4$ as $- 1 (4)$ .

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

Factor $- 1$ out of $- (x) - 1 (4)$ .

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

Rewrite $- (x + 4)$ as $- 1 (x + 4)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{- 1 (2 (x - 7))}{5 x}$

Step 8

Simplify the expression.

Multiply $- 1$ by $2$ .

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

Move the negative in front of the fraction.

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

Simplify ((-x^2+x+20)/(5x^2-25x))÷((x+4)/(2x-14))

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