1. **State the problem:** Find the domain and simplify the multiplication of the expressions: $$\frac{bx + 3b}{x^2 - 25} \times \frac{(x - 5)^2}{ax + 3a}$$ 2. **Identify the domain restrictions:** The denominators cannot be zero. - For $x^2 - 25 = 0$, solve $x^2 = 25$ so $x = \pm 5$. - For $ax + 3a = 0$, factor out $a$: $a(x + 3) = 0$. Since $a \neq 0$, $x + 3 = 0$ so $x = -3$. Thus, the domain restrictions are: $$x \neq 5, \quad x \neq -5, \quad x \neq -3$$ 3. **Rewrite and factor expressions:** - Numerator 1: $bx + 3b = b(x + 3)$ - Denominator 1: $x^2 - 25 = (x - 5)(x + 5)$ - Numerator 2: $(x - 5)^2$ - Denominator 2: $ax + 3a = a(x + 3)$ 4. **Multiply the fractions:** $$\frac{b(x + 3)}{(x - 5)(x + 5)} \times \frac{(x - 5)^2}{a(x + 3)} = \frac{b(x + 3)(x - 5)^2}{(x - 5)(x + 5) a (x + 3)}$$ 5. **Cancel common factors:** - Cancel $(x + 3)$ numerator and denominator: $$\frac{b\cancel{(x + 3)}(x - 5)^2}{(x - 5)(x + 5) a \cancel{(x + 3)}}$$ - Cancel one $(x - 5)$ from numerator and denominator: $$\frac{b \cancel{(x - 5)} (x - 5)}{\cancel{(x - 5)} (x + 5) a}$$ 6. **Simplified expression:** $$\frac{b(x - 5)}{a(x + 5)}$$ 7. **Summary:** - Domain: $x \neq 5, x \neq -5, x \neq -3$ - Simplified expression: $$\frac{b(x - 5)}{a(x + 5)}$$