1. **Problem statement:** We are given the expression $3x^2 - 4ax - 4a^2$ which is exactly divisible by $x + 2$. We need to find the value of $a$. 2. **Key concept:** If a polynomial $P(x)$ is exactly divisible by $(x - r)$, then $P(r) = 0$ (Remainder Theorem). 3. Here, the divisor is $x + 2$, which can be written as $x - (-2)$. So, $r = -2$. 4. Substitute $x = -2$ into the polynomial and set it equal to zero: $$3(-2)^2 - 4a(-2) - 4a^2 = 0$$ 5. Simplify step-by-step: $$3 \times 4 + 8a - 4a^2 = 0$$ $$12 + 8a - 4a^2 = 0$$ 6. Rearrange the equation: $$-4a^2 + 8a + 12 = 0$$ 7. Divide the entire equation by $-4$ to simplify: $$a^2 - 2a - 3 = 0$$ 8. Factorize the quadratic: $$(a - 3)(a + 1) = 0$$ 9. Solve for $a$: $$a - 3 = 0 \Rightarrow a = 3$$ $$a + 1 = 0 \Rightarrow a = -1$$ **Final answer:** The values of $a$ are $3$ or $-1$.