Question

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

\\\frac{2x^2 + 7x - 30}{4x^2 - 25} \div \frac{x^2 + 14x + 49}{x^2 + 4x - 21}\\

\\\frac{(x + 6)(x - ?)}{(2x + \quad)(x + 7)}\\

Explanation:

Factor each polynomial

Using the Factoring Trinomials knowledge point

$$
LATEXBLOCK0
$$

Rewrite division as multiplication

Using the Dividing Rational Expressions knowledge point

$$ \frac{2x^2 + 7x - 30}{4x^2 - 25} \div \frac{x^2 + 14x + 49}{x^2 + 4x - 21} = \frac{(2x - 5)(x + 6)}{(2x - 5)(2x + 5)} \cdot \frac{(x + 7)(x - 3)}{(x + 7)^2} $$

Simplify the expression

Using the Simplifying Rational Expressions knowledge point

$$
LATEXBLOCK1
$$

Identify the missing values

Comparing the simplified quotient to the target template:

$$ \frac{(x + 6)(x - [ ? ])}{(2x + [ \quad ])(x + 7)} = \frac{(x + 6)(x - 3)}{(2x + 5)(x + 7)} $$

Thus, the value in the green box with the question mark \([ ? ]\) is \(3\), and the value in the blue box \([ \quad ]\) is \(5\).

Answer:

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

$$\frac{2x^2 + 7x - 30}{4x^2 - 25} \div \frac{x^2 + 14x + 49}{x^2 + 4x - 21}$$

The simplified quotient is:

$$\frac{(x + 6)(x - \text{3})}{(2x + \text{5})(x + 7)}$$