1. **State the problem:** Simplify the rational expression $$\frac{20 - 5v^2}{v^2 - 7v + 10}$$. 2. **Recall the formula and rules:** To simplify a rational expression, factor both numerator and denominator completely and then cancel any common factors. 3. **Factor the numerator:** $$20 - 5v^2 = 5(4 - v^2)$$ Recognize that $$4 - v^2$$ is a difference of squares: $$4 - v^2 = (2 - v)(2 + v)$$ So, $$20 - 5v^2 = 5(2 - v)(2 + v)$$ 4. **Factor the denominator:** $$v^2 - 7v + 10$$ Find two numbers that multiply to 10 and add to -7: -5 and -2. So, $$v^2 - 7v + 10 = (v - 5)(v - 2)$$ 5. **Rewrite the expression:** $$\frac{5(2 - v)(2 + v)}{(v - 5)(v - 2)}$$ 6. **Simplify common factors:** Notice that $$2 - v = -(v - 2)$$, so: $$5(2 - v)(2 + v) = 5[-(v - 2)](2 + v) = -5(v - 2)(2 + v)$$ 7. Substitute back: $$\frac{-5(v - 2)(2 + v)}{(v - 5)(v - 2)}$$ 8. Cancel the common factor $$v - 2$$: $$\frac{-5\cancel{(v - 2)}(2 + v)}{(v - 5)\cancel{(v - 2)}} = \frac{-5(2 + v)}{v - 5}$$ 9. **Final simplified form:** $$\boxed{\frac{-5(2 + v)}{v - 5}}$$ This is the simplified form of the original rational expression.