Simplify ((5x^2-19x+12)/(10x-30))÷(16-25x^2)

$\frac{5 x^{2} - 19 x + 12}{10 x - 30} \div (16 - 25 x^{2})$

Rewrite the division as a fraction.

$\frac{\frac{5 x^{2} - 19 x + 12}{10 x - 30}}{16 - 25 x^{2}}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{5 x^{2} - 19 x + 12}{10 x - 30} \cdot \frac{1}{16 - 25 x^{2}}$

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 $a \cdot c = 5 \cdot 12 = 60$ and whose sum is $b = - 19$ .

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

$\frac{5 x^{2} - 19 (x) + 12}{10 x - 30} \cdot \frac{1}{16 - 25 x^{2}}$

Rewrite $- 19$ as $- 4$ plus $- 15$

$\frac{5 x^{2} + (- 4 - 15) x + 12}{10 x - 30} \cdot \frac{1}{16 - 25 x^{2}}$

Apply the distributive property.

$\frac{5 x^{2} - 4 x - 15 x + 12}{10 x - 30} \cdot \frac{1}{16 - 25 x^{2}}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(5 x^{2} - 4 x) - 15 x + 12}{10 x - 30} \cdot \frac{1}{16 - 25 x^{2}}$

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

$\frac{x (5 x - 4) - 3 (5 x - 4)}{10 x - 30} \cdot \frac{1}{16 - 25 x^{2}}$

Factor the polynomial by factoring out the greatest common factor, $5 x - 4$ .

$\frac{(5 x - 4) (x - 3)}{10 x - 30} \cdot \frac{1}{16 - 25 x^{2}}$

Simplify terms.

Factor $10$ out of $10 x - 30$ .

Factor $10$ out of $10 x$ .

$\frac{(5 x - 4) (x - 3)}{10 (x) - 30} \cdot \frac{1}{16 - 25 x^{2}}$

Factor $10$ out of $- 30$ .

$\frac{(5 x - 4) (x - 3)}{10 x + 10 \cdot - 3} \cdot \frac{1}{16 - 25 x^{2}}$

Factor $10$ out of $10 x + 10 \cdot - 3$ .

$\frac{(5 x - 4) (x - 3)}{10 (x - 3)} \cdot \frac{1}{16 - 25 x^{2}}$

Cancel the common factor of $x - 3$ .

Cancel the common factor.

$\frac{(5 x - 4) (x - 3)}{10 (x - 3)} \cdot \frac{1}{16 - 25 x^{2}}$

Rewrite the expression.

$\frac{5 x - 4}{10} \cdot \frac{1}{16 - 25 x^{2}}$

Simplify the denominator.

Rewrite $16$ as $4^{2}$ .

$\frac{5 x - 4}{10} \cdot \frac{1}{4^{2} - 25 x^{2}}$

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

$\frac{5 x - 4}{10} \cdot \frac{1}{4^{2} - {(5 x)}^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = 5 x$ .

$\frac{5 x - 4}{10} \cdot \frac{1}{(4 + 5 x) (4 - (5 x))}$

Multiply $5$ by $- 1$ .

$\frac{5 x - 4}{10} \cdot \frac{1}{(4 + 5 x) (4 - 5 x)}$

Simplify terms.

Multiply $\frac{5 x - 4}{10}$ by $\frac{1}{(4 + 5 x) (4 - 5 x)}$ .

$\frac{5 x - 4}{10 ((4 + 5 x) (4 - 5 x))}$

Cancel the common factor of $5 x - 4$ and $4 - 5 x$ .

Factor $- 1$ out of $5 x$ .

$\frac{- (- 5 x) - 4}{10 ((4 + 5 x) (4 - 5 x))}$

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

$\frac{- (- 5 x) - 1 (4)}{10 ((4 + 5 x) (4 - 5 x))}$

Factor $- 1$ out of $- (- 5 x) - 1 (4)$ .

$\frac{- (- 5 x + 4)}{10 ((4 + 5 x) (4 - 5 x))}$

Rewrite $- (- 5 x + 4)$ as $- 1 (- 5 x + 4)$ .

$\frac{- 1 (- 5 x + 4)}{10 ((4 + 5 x) (4 - 5 x))}$

Reorder terms.

$\frac{- 1 (- 5 x + 4)}{10 (4 + 5 x) (- 5 x + 4)}$

Cancel the common factor.

$\frac{- 1 (- 5 x + 4)}{10 (4 + 5 x) (- 5 x + 4)}$

Rewrite the expression.

$\frac{- 1}{10 (4 + 5 x)}$

Move the negative in front of the fraction.

$- \frac{1}{10 (4 + 5 x)}$

Simplify ((5x^2-19x+12)/(10x-30))÷(16-25x^2)

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