1. **State the problem:** Given the polynomial function $$f(x) = -10 \left(x + \frac{4}{5}\right)^2 (x + 4)^3,$$ we need to find: (a) The real zeros and their multiplicities. (b) Whether the graph crosses or touches the x-axis at each zero. (c) The maximum number of turning points. (d) The end behavior of the graph. 2. **Find the real zeros and multiplicities:** The zeros occur where each factor equals zero: - For $$x + \frac{4}{5} = 0,$$ we get $$x = -\frac{4}{5}$$ with multiplicity 2. - For $$x + 4 = 0,$$ we get $$x = -4$$ with multiplicity 3. 3. **Determine crossing or touching at each zero:** - At $$x = -\frac{4}{5}$$, multiplicity 2 (even), the graph **touches** the x-axis and turns around. - At $$x = -4$$, multiplicity 3 (odd), the graph **crosses** the x-axis. 4. **Maximum number of turning points:** The degree of the polynomial is the sum of multiplicities: $$2 + 3 = 5.$$ The maximum number of turning points is one less than the degree: $$5 - 1 = 4.$$ 5. **End behavior:** Expand the leading term by multiplying the leading terms of each factor: - Leading term of $$\left(x + \frac{4}{5}\right)^2$$ is $$x^2$$. - Leading term of $$(x + 4)^3$$ is $$x^3$$. - Multiply by $$-10$$ gives: $$-10 \times x^2 \times x^3 = -10 x^5.$$ For large $$|x|$$, $$f(x)$$ behaves like $$-10 x^5$$. **Final answers:** (a) Real zeros: $$x = -\frac{4}{5}$$ (multiplicity 2), $$x = -4$$ (multiplicity 3). (b) At $$x = -\frac{4}{5}$$, graph touches the x-axis; at $$x = -4$$, graph crosses the x-axis. (c) Maximum turning points: 4. (d) End behavior resembles $$-10 x^5$$ for large $$|x|$$.