1. **State the problem:** We need to sketch the curve of the function $$y = (x+3)(x+1)(x-2)$$. 2. **Formula and important rules:** This is a cubic polynomial function. To sketch it, we find its roots (where $$y=0$$), analyze its end behavior, and find critical points (where the slope is zero) to understand its shape. 3. **Find the roots:** Set $$y=0$$: $$ (x+3)(x+1)(x-2) = 0 $$ Roots are $$x = -3, -1, 2$$. 4. **End behavior:** Since the leading term when expanded is $$x^3$$ (positive coefficient), as $$x \to \infty$$, $$y \to \infty$$, and as $$x \to -\infty$$, $$y \to -\infty$$. 5. **Find critical points:** Differentiate: $$y = (x+3)(x+1)(x-2)$$ Use product rule: $$y' = (x+1)(x-2) + (x+3)(x-2) + (x+3)(x+1)$$ Simplify each: $$(x+1)(x-2) = x^2 - x - 2$$ $$(x+3)(x-2) = x^2 + x - 6$$ $$(x+3)(x+1) = x^2 + 4x + 3$$ Sum: $$y' = (x^2 - x - 2) + (x^2 + x - 6) + (x^2 + 4x + 3) = 3x^2 + 4x - 5$$ 6. **Solve for critical points:** $$3x^2 + 4x - 5 = 0$$ Use quadratic formula: $$x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 3 \cdot (-5)}}{2 \cdot 3} = \frac{-4 \pm \sqrt{16 + 60}}{6} = \frac{-4 \pm \sqrt{76}}{6}$$ $$\sqrt{76} \approx 8.7178$$ So, $$x_1 = \frac{-4 - 8.7178}{6} \approx -2.7863$$ $$x_2 = \frac{-4 + 8.7178}{6} \approx 0.7863$$ 7. **Find corresponding $$y$$ values:** $$y(-2.7863) = (-2.7863+3)(-2.7863+1)(-2.7863-2) \approx (0.2137)(-1.7863)(-4.7863) \approx 1.83$$ $$y(0.7863) = (0.7863+3)(0.7863+1)(0.7863-2) \approx (3.7863)(1.7863)(-1.2137) \approx -8.21$$ 8. **Summary:** - Roots at $$x = -3, -1, 2$$ - Local maximum near $$(-2.79, 1.83)$$ - Local minimum near $$(0.79, -8.21)$$ - End behavior: $$y \to -\infty$$ as $$x \to -\infty$$ and $$y \to \infty$$ as $$x \to \infty$$ This information allows us to sketch the curve accurately.