13) Perform each product or quotient of rational expressions. \[ \text { A: } \frac{x^{2}+x-2}{x^{2}-2 x-3} \cdot \frac{x+1}{4 x+8} \]

We begin with the expression: \[ \frac{x^{2}+x-2}{x^{2}-2x-3} \cdot \frac{x+1}{4x+8}. \] **Step 1. Factor each polynomial** Factor the quadratic in the numerator: \[ x^{2}+x-2 = (x+2)(x-1). \] Factor the quadratic in the denominator: \[ x^{2}-2x-3 = (x-3)(x+1). \] Factor the linear expression in the denominator of the second fraction: \[ 4x+8 = 4(x+2). \] Now, the expression becomes: \[ \frac{(x+2)(x-1)}{(x-3)(x+1)} \cdot \frac{x+1}{4(x+2)}. \] **Step 2. Cancel common factors** Notice that the factor \((x+2)\) appears in both the numerator of the first fraction and the denominator of the second fraction, and the factor \((x+1)\) appears in both the denominator of the first fraction and the numerator of the second fraction. Cancel these common factors: \[ \frac{\cancel{(x+2)}(x-1)}{(x-3)\cancel{(x+1)}} \cdot \frac{\cancel{(x+1)}}{4\cancel{(x+2)}} = \frac{x-1}{4(x-3)}. \] **Final Answer** \[ \boxed{\frac{x-1}{4(x-3)}.} \]