1. **State the problem:** Simplify the expression $$\frac{2b - 5}{b^2 - 2b - 15} + \frac{3b}{b^2 + b - 30} \times \frac{b^2 + 8b + 12}{b + 3}$$ 2. **Factor all polynomials:** - Factor denominator $b^2 - 2b - 15$: $$b^2 - 2b - 15 = (b - 5)(b + 3)$$ - Factor denominator $b^2 + b - 30$: $$b^2 + b - 30 = (b + 6)(b - 5)$$ - Factor numerator $b^2 + 8b + 12$: $$b^2 + 8b + 12 = (b + 6)(b + 2)$$ 3. **Rewrite the expression with factored forms:** $$\frac{2b - 5}{(b - 5)(b + 3)} + \frac{3b}{(b + 6)(b - 5)} \times \frac{(b + 6)(b + 2)}{b + 3}$$ 4. **Simplify the multiplication part:** $$\frac{3b}{(b + 6)(b - 5)} \times \frac{(b + 6)(b + 2)}{b + 3} = \frac{3b \cancel{(b + 6)} (b + 2)}{\cancel{(b + 6)} (b - 5)(b + 3)} = \frac{3b (b + 2)}{(b - 5)(b + 3)}$$ 5. **Rewrite the entire expression:** $$\frac{2b - 5}{(b - 5)(b + 3)} + \frac{3b (b + 2)}{(b - 5)(b + 3)}$$ 6. **Combine the fractions since they have the same denominator:** $$\frac{2b - 5 + 3b (b + 2)}{(b - 5)(b + 3)}$$ 7. **Expand the numerator:** $$3b (b + 2) = 3b^2 + 6b$$ So numerator becomes: $$2b - 5 + 3b^2 + 6b = 3b^2 + 8b - 5$$ 8. **Final expression:** $$\frac{3b^2 + 8b - 5}{(b - 5)(b + 3)}$$ 9. **Check if numerator factors:** Discriminant $\Delta = 8^2 - 4 \times 3 \times (-5) = 64 + 60 = 124$ which is not a perfect square, so numerator does not factor nicely. **Answer:** $$\boxed{\frac{3b^2 + 8b - 5}{(b - 5)(b + 3)}}$$