1. **State the problem:** Simplify and analyze the quadratic expression $-3x^2 + 12x$. 2. **Formula and rules:** This is a quadratic expression of the form $ax^2 + bx + c$ where $a = -3$, $b = 12$, and $c = 0$. 3. **Factor the expression:** Factor out the greatest common factor (GCF) which is $-3x$: $$-3x^2 + 12x = -3x(x - 4)$$ 4. **Find the roots (zeros):** Set the expression equal to zero: $$-3x(x - 4) = 0$$ This implies either $-3x = 0$ or $x - 4 = 0$, so $$x = 0 \quad \text{or} \quad x = 4$$ 5. **Vertex and axis of symmetry:** The vertex of a parabola $y = ax^2 + bx + c$ is at $$x = -\frac{b}{2a} = -\frac{12}{2 \times (-3)} = 2$$ Calculate $y$ at $x=2$: $$y = -3(2)^2 + 12(2) = -3(4) + 24 = -12 + 24 = 12$$ So the vertex is at $(2, 12)$. 6. **Interpretation:** The parabola opens downward (since $a = -3 < 0$), has roots at $0$ and $4$, and a maximum point at $(2, 12)$. **Final answer:** $$-3x^2 + 12x = -3x(x - 4)$$ Roots: $x=0$, $x=4$ Vertex: $(2, 12)$ Parabola opens downward.