1. **State the problem:** We need to find the multiplicity of the zero $x=4$ in the function $h(x) = f(x) \cdot g(x)$, where $$f(x) = 3x^2 (x + 4) (x + 7)^4 (x - 4)^3$$ $$g(x) = 3 (x + 7)^2 (x + 3)^2 (x + 4) (x - 8)^2$$ 2. **Recall the rule for multiplicity:** The multiplicity of a zero $x=a$ in a product of functions is the sum of the multiplicities of $x=a$ in each function. 3. **Find the multiplicity of $x=4$ in $f(x)$:** - In $f(x)$, the factor $(x - 4)^3$ shows that $x=4$ is a zero of multiplicity 3. 4. **Find the multiplicity of $x=4$ in $g(x)$:** - In $g(x)$, the factor $(x - 4)$ does not appear, so the multiplicity of $x=4$ in $g(x)$ is 0. 5. **Add the multiplicities:** $$\text{Multiplicity of } x=4 \text{ in } h(x) = 3 + 0 = 3$$ **Final answer:** The multiplicity of the zero $x=4$ in $h(x)$ is **3**.