1. **State the problem:** We are given the function $y = 3(x + 5)(x - 2)$ and want to understand its properties. 2. **Formula and rules:** This is a quadratic function in factored form. To analyze it, we can expand it to standard form and find intercepts and vertex. 3. **Expand the expression:** $$y = 3(x + 5)(x - 2) = 3\left(x^2 - 2x + 5x - 10\right) = 3(x^2 + 3x - 10)$$ 4. **Distribute the 3:** $$y = 3x^2 + 9x - 30$$ 5. **Find x-intercepts:** Set $y=0$: $$0 = 3(x + 5)(x - 2)$$ This implies either $x + 5 = 0$ or $x - 2 = 0$, so $$x = -5 \quad \text{or} \quad x = 2$$ 6. **Find y-intercept:** Set $x=0$: $$y = 3(0 + 5)(0 - 2) = 3(5)(-2) = -30$$ So the y-intercept is $(0, -30)$. 7. **Find vertex:** The vertex of a parabola $y = ax^2 + bx + c$ is at $$x = -\frac{b}{2a} = -\frac{9}{2 \times 3} = -\frac{9}{6} = -1.5$$ Calculate $y$ at $x = -1.5$: $$y = 3(-1.5)^2 + 9(-1.5) - 30 = 3(2.25) - 13.5 - 30 = 6.75 - 13.5 - 30 = -36.75$$ So the vertex is at $(-1.5, -36.75)$. **Final answer:** The quadratic function $y = 3(x + 5)(x - 2)$ has x-intercepts at $x = -5$ and $x = 2$, y-intercept at $(0, -30)$, and vertex at $(-1.5, -36.75)$.