Questions: Divide as indicated. (x^2-4)/(5x-5) ÷ (x^2+4x+4)/(x^2+x-2) 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. (x^2-4)/(5x-5) ÷ (x^2+4x+4)/(x^2+x-2)=□, x ≠ □ B. (x^2-4)/(5x-5) ÷ (x^2+4x+4)/(x^2+x-2)=□, no numbers must be excluded.

Solution Steps

To solve the given division of rational expressions, we need to follow these steps:

1. Factorize all the polynomials in the numerator and the denominator.
2. Rewrite the division as multiplication by the reciprocal of the second fraction.
3. Simplify the resulting expression by canceling out common factors.
4. Identify any values of \( x \) that must be excluded to avoid division by zero.

First, we factorize all the polynomials in the numerator and the denominator: \[ \text{Numerator 1: } x^2 - 4 = (x - 2)(x + 2) \] \[ \text{Denominator 1: } 5x - 5 = 5(x - 1) \] \[ \text{Numerator 2: } x^2 + 4x + 4 = (x + 2)^2 \] \[ \text{Denominator 2: } x^2 + x - 2 = (x - 1)(x + 2) \]

Rewrite the division as multiplication by the reciprocal of the second fraction: \[ \frac{(x - 2)(x + 2)}{5(x - 1)} \div \frac{(x + 2)^2}{(x - 1)(x + 2)} = \frac{(x - 2)(x + 2)}{5(x - 1)} \times \frac{(x - 1)(x + 2)}{(x + 2)^2} \]

Simplify the resulting expression by canceling out common factors: \[ \frac{(x - 2)(x + 2)}{5(x - 1)} \times \frac{(x - 1)(x + 2)}{(x + 2)^2} = \frac{(x - 2) \cancel{(x + 2)}}{5 \cancel{(x - 1)}} \times \frac{\cancel{(x - 1)} \cancel{(x + 2)}}{\cancel{(x + 2)} \cancel{(x + 2)}} = \frac{x - 2}{5} \]

Identify the values of \( x \) that must be excluded to avoid division by zero: \[ 5(x - 1) = 0 \implies x = 1 \] \[ (x - 1)(x + 2) = 0 \implies x = 1, -2 \] \[ (x + 2)^2 = 0 \implies x = -2 \] Thus, the excluded values are \( x = 1 \) and \( x = -2 \).

Final Answer