1. **State the problem:** We have a function $$f(x) = a(x + 2)(x - a)(x - 8)$$ where $$a$$ is a constant. We want to find which value of $$a$$ makes $$f(2.5)$$ negative. 2. **Substitute $$x = 2.5$$ into the function:** $$f(2.5) = a(2.5 + 2)(2.5 - a)(2.5 - 8)$$ 3. **Simplify the factors:** $$2.5 + 2 = 4.5$$ $$2.5 - 8 = -5.5$$ So, $$f(2.5) = a \times 4.5 \times (2.5 - a) \times (-5.5)$$ 4. **Rewrite the expression:** $$f(2.5) = a \times 4.5 \times (2.5 - a) \times (-5.5) = -24.75 a (2.5 - a)$$ 5. **Analyze the sign of $$f(2.5)$$:** We want $$f(2.5) < 0$$, so: $$-24.75 a (2.5 - a) < 0$$ Since $$-24.75$$ is negative, multiply both sides by $$-1$$ and reverse inequality: $$a (2.5 - a) > 0$$ 6. **Solve inequality $$a (2.5 - a) > 0$$:** This product is positive if both factors are positive or both are negative. - Case 1: $$a > 0$$ and $$2.5 - a > 0 \Rightarrow a < 2.5$$ - Case 2: $$a < 0$$ and $$2.5 - a < 0 \Rightarrow a > 2.5$$ (impossible) So the solution is: $$0 < a < 2.5$$ 7. **Check the options:** - a = 4 (not in interval) - a = 2 (in interval) - a = -2 (not in interval) - a = 0 (not in interval) **Answer:** The value of $$a$$ that makes $$f(2.5)$$ negative is $$2$$.