1. **State the problem:** We need to find the value of $a$ given the function $f(x) = ax^2 + 2x$ and the condition $f(3) = 12$. 2. **Write the formula and substitute:** The function is $f(x) = ax^2 + 2x$. Substitute $x = 3$ and $f(3) = 12$: $$12 = a(3)^2 + 2(3)$$ 3. **Simplify the equation:** $$12 = 9a + 6$$ 4. **Isolate $a$:** $$12 - 6 = 9a$$ $$6 = 9a$$ 5. **Divide both sides by 9:** $$\frac{6}{\cancel{9}} = a \times \frac{\cancel{9}}{9}$$ $$a = \frac{6}{9}$$ 6. **Simplify the fraction:** $$a = \frac{2}{3}$$ **Final answer:** $a = \frac{2}{3}$