1. **State the problem:** Find the domain and simplify the multiplication of the rational expressions $$\frac{bx + 3b}{x^2 - 25} \cdot \frac{(x - 5)^2}{ax + 3a}$$ 2. **Identify the domain restrictions:** The domain excludes values that make any denominator zero. - Denominator 1: $x^2 - 25 = 0 \Rightarrow (x-5)(x+5) = 0 \Rightarrow x = 5, -5$ - Denominator 2: $ax + 3a = 0 \Rightarrow a(x + 3) = 0$; assuming $a \neq 0$, then $x = -3$ **Domain:** $x \neq -5, 5, -3$ 3. **Simplify the expression:** Write the expression: $$\frac{b(x + 3)}{(x-5)(x+5)} \cdot \frac{(x-5)^2}{a(x+3)}$$ 4. **Multiply numerators and denominators:** $$\frac{b(x + 3)(x-5)^2}{a(x+3)(x-5)(x+5)}$$ 5. **Cancel common factors:** Cancel $(x+3)$ and one $(x-5)$: $$\frac{b\cancel{(x + 3)}(x-5)\cancel{(x-5)}}{a\cancel{(x+3)}\cancel{(x-5)}(x+5)} = \frac{b(x-5)}{a(x+5)}$$ 6. **Final simplified expression:** $$\frac{b(x-5)}{a(x+5)}$$ 7. **Summary:** - Domain: $x \neq -5, 5, -3$ - Simplified expression: $\frac{b(x-5)}{a(x+5)}$