1. **State the problem:** We are given the quadratic function $f(x) = 3(x + 2)(x + 4)$ and want to understand its shape and key features. 2. **Formula and explanation:** The function is in factored form, which shows the roots directly. The roots are the values of $x$ that make each factor zero: $$x + 2 = 0 \Rightarrow x = -2$$ $$x + 4 = 0 \Rightarrow x = -4$$ 3. **Expand the function:** To better understand the shape, expand the product: $$f(x) = 3(x + 2)(x + 4) = 3(x^2 + 6x + 8) = 3x^2 + 18x + 24$$ 4. **Find the vertex:** The vertex of a quadratic $ax^2 + bx + c$ is at $$x = -\frac{b}{2a} = -\frac{18}{2 \times 3} = -3$$ 5. **Calculate the vertex's y-value:** Substitute $x = -3$ into $f(x)$: $$f(-3) = 3(-3 + 2)(-3 + 4) = 3(-1)(1) = -3$$ 6. **Interpretation:** The parabola opens upwards (since $a=3 > 0$), has roots at $x = -4$ and $x = -2$, and a vertex at $(-3, -3)$ which is the minimum point. **Final answer:** The quadratic function $f(x) = 3(x + 2)(x + 4)$ has roots at $x = -4$ and $x = -2$, and a vertex at $(-3, -3)$ with the parabola opening upwards.