Questions: Divide as indicated. (12x^2+16)/(x-5) ÷ (15x^2+20)/(x^2-25) Select the correct choice below and fill in the answer box(es) to complete your choice. (Simplify your answer. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed) A. (12x^2+16)/(x-5)+(15x^2+20)/(x^2-25)=, x ≠ B. (12x^2+16)/(x-5) ÷ (15x^2+20)/(x^2-25) = no numbers must be excluded.

$Divide as indicated. (12x^2+16)/(x-5) ÷ (15x^2+20)/(x^2-25) Select the correct choice below and fill in the answer box(es) to complete your choice. (Simplify your answer. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed) A. (12x^2+16)/(x-5)+(15x^2+20)/(x^2-25)=, x ≠ B. (12x^2+16)/(x-5) ÷ (15x^2+20)/(x^2-25) = no numbers must be excluded.$

Transcript text: Divide as indicated. \[ \frac{12 x^{2}+16}{x-5} \div \frac{15 x^{2}+20}{x^{2}-25} \] Select the correct choice below and fill in the answer box(es) to complete your choice. (Simplify your answer. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed) A. $\frac{12 x^{2}+16}{x-5}+\frac{15 x^{2}+20}{x^{2}-25}=\square, x \neq \square$ B. $\frac{12 x^{2}+16}{x-5} \div \frac{15 x^{2}+20}{x^{2}-25}=$ $\square$ no numbers must be excluded.

Solution

Solution Steps

To solve the given problem, we need to divide two rational expressions. The division of fractions can be converted into multiplication by taking the reciprocal of the divisor. After that, we simplify the resulting expression by factoring and canceling common terms.

Solution Approach

Convert the division into multiplication by taking the reciprocal of the second fraction.
Factorize the numerators and denominators of both fractions.
Simplify the resulting expression by canceling out common factors.
Identify any restrictions on the variable $ x $ to avoid division by zero.

Step 1: Convert Division to Multiplication by Reciprocal

To divide the rational expressions, we first convert the division into multiplication by taking the reciprocal of the second fraction: \[ \frac{12x^2 + 16}{x - 5} \div \frac{15x^2 + 20}{x^2 - 25} = \frac{12x^2 + 16}{x - 5} \times \frac{x^2 - 25}{15x^2 + 20} \]

Step 2: Factorize the Expressions

Next, we factorize the numerators and denominators: \[ 12x^2 + 16 = 4(3x^2 + 4) \] \[ x^2 - 25 = (x - 5)(x + 5) \] \[ 15x^2 + 20 = 5(3x^2 + 4) \]

Step 3: Combine and Simplify

We then combine the expressions and simplify by canceling common factors: \[ \frac{4(3x^2 + 4) \cdot (x - 5)(x + 5)}{(x - 5) \cdot 5(3x^2 + 4)} = \frac{4(x + 5)}{5} \]

Step 4: Identify Restrictions

To avoid division by zero, we identify the restrictions on $ x $: \[ x \neq 5, \quad x^2 - 25 \neq 0 \implies x \neq \pm 5 \]