Question

use factoring to find the simplified quotient of these rational expressions.

\\\frac{3x^2 + 13x - 30}{x^2 + 12x + 36} \div \frac{9x^2 - 25}{3x^2 - 5x - 12}\\

\\\frac{(3x + 4)(x - ? )}{(x + \quad)(3x + 5)}\\

Explanation:

Factor each polynomial in the expression

$$
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$$

Rewrite the division as multiplication by the reciprocal

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

Simplify the expression by canceling common factors

$$ \frac{\cancel{(3x - 5)}\cancel{(x + 6)}}{(x + 6)^{\cancel{2}}} \cdot \frac{(3x + 4)(x - 3)}{\cancel{(3x - 5)}(3x + 5)} = \frac{(3x + 4)(x - 3)}{(x + 6)(3x + 5)} $$

Answer:

Use factoring to find the simplified quotient of these rational expressions.

$$\frac{3x^2 + 13x - 30}{x^2 + 12x + 36} \div \frac{9x^2 - 25}{3x^2 - 5x - 12}$$

The simplified quotient is <blank>\(\frac{(3x + 4)(x - 3)}{(x + 6)(3x + 5)}\)</blank>.