1. **State the problem:** Simplify the rational expression $$\frac{75 - 3v^2}{v^2 + v - 30}$$. 2. **Factor the numerator and denominator:** - Numerator: $$75 - 3v^2 = 3(25 - v^2)$$. - Recognize that $$25 - v^2$$ is a difference of squares: $$25 - v^2 = (5 - v)(5 + v)$$. - So numerator becomes $$3(5 - v)(5 + v)$$. - Denominator: $$v^2 + v - 30$$. - Find two numbers that multiply to $$-30$$ and add to $$1$$: these are $$6$$ and $$-5$$. - So denominator factors as $$(v + 6)(v - 5)$$. 3. **Rewrite the expression:** $$\frac{3(5 - v)(5 + v)}{(v + 6)(v - 5)}$$. 4. **Simplify by canceling common factors:** - Notice that $$5 - v = -(v - 5)$$. - So numerator can be written as $$3(-(v - 5))(5 + v) = -3(v - 5)(5 + v)$$. - The expression is now: $$\frac{-3(v - 5)(5 + v)}{(v + 6)(v - 5)}$$. - Cancel the common factor $$(v - 5)$$: $$\frac{-3\cancel{(v - 5)}(5 + v)}{(v + 6)\cancel{(v - 5)}}$$. 5. **Final simplified expression:** $$\frac{-3(5 + v)}{v + 6}$$. 6. **Optional:** Rewrite numerator as $$-3(v + 5)$$ for clarity. **Answer:** $$\boxed{\frac{-3(v + 5)}{v + 6}}$$