1. **State the problem:** We need to analyze the function $$y = (x+1)^2 (x-4)^3$$. 2. **Formula and rules:** This is a polynomial function expressed as a product of powers of binomials. To understand its behavior, we look at intercepts, multiplicities, and end behavior. 3. **Find the x-intercepts:** Set $$y=0$$: $$ (x+1)^2 (x-4)^3 = 0 $$ This implies $$x+1=0$$ or $$x-4=0$$. 4. **Solve for intercepts:** - $$x=-1$$ with multiplicity 2 (even multiplicity means the graph touches the x-axis and turns around). - $$x=4$$ with multiplicity 3 (odd multiplicity means the graph crosses the x-axis and flattens). 5. **Find the y-intercept:** Set $$x=0$$: $$y = (0+1)^2 (0-4)^3 = 1^2 \times (-4)^3 = 1 \times (-64) = -64$$. 6. **End behavior:** The degree is $$2 + 3 = 5$$ (odd degree), and the leading coefficient is positive (from expansion, leading term is $$x^5$$). So as $$x \to \infty$$, $$y \to \infty$$, and as $$x \to -\infty$$, $$y \to -\infty$$. **Final answer:** The function $$y = (x+1)^2 (x-4)^3$$ has x-intercepts at $$x=-1$$ (touches axis) and $$x=4$$ (crosses axis), y-intercept at $$-64$$, and end behavior rising to infinity on the right and falling to negative infinity on the left.