Questions: Perform the indicated operations and simplify the result. Leave the answer in factored form. (x^2-9)/(4x+12) * (x+2)/(x-3) A. (x+2)/4 B. x+2 C. ((x+2)(x-3))/(4(x+3)) D. 1/4

Solution Steps

To solve the given expression, we need to factorize the numerator and the denominator where possible, then simplify the expression by canceling out common factors.

1. Factorize \(x^2 - 9\) as \((x - 3)(x + 3)\).
2. Factorize \(4x + 12\) as \(4(x + 3)\).
3. Substitute these factorizations into the original expression.
4. Simplify by canceling out common factors.

First, we factorize the numerator and the denominator of the given expression: \[ \frac{x^2 - 9}{4x + 12} \cdot \frac{x + 2}{x - 3} \]

Factorize \(x^2 - 9\) as \((x - 3)(x + 3)\): \[ x^2 - 9 = (x - 3)(x + 3) \]

Factorize \(4x + 12\) as \(4(x + 3)\): \[ 4x + 12 = 4(x + 3) \]

Substitute the factored forms into the original expression: \[ \frac{(x - 3)(x + 3)}{4(x + 3)} \cdot \frac{x + 2}{x - 3} \]

Cancel out the common factors \((x - 3)\) and \((x + 3)\) from the numerator and the denominator: \[ \frac{(x - 3)(x + 3)}{4(x + 3)} \cdot \frac{x + 2}{x - 3} = \frac{(x + 2)}{4} \]

Final Answer