1. **State the problem:** We need to find a polynomial function $m(x)$ of degree 4 with zeros at $0$, $5$, and $3 - 2i$. 2. **Recall the important rule:** For polynomials with real coefficients, complex roots come in conjugate pairs. Since $3 - 2i$ is a root, its conjugate $3 + 2i$ must also be a root. 3. **List all roots:** $0$, $5$, $3 - 2i$, and $3 + 2i$. 4. **Write the factors:** Each root corresponds to a factor: $$x, (x - 5), (x - (3 - 2i)), (x - (3 + 2i))$$ 5. **Multiply the complex conjugate factors:** $$ (x - (3 - 2i))(x - (3 + 2i)) = (x - 3 + 2i)(x - 3 - 2i) $$ Use the difference of squares formula: $$ = ((x - 3))^2 - (2i)^2 = (x - 3)^2 - (-4) = (x - 3)^2 + 4 $$ 6. **Expand $(x - 3)^2 + 4$:** $$ (x - 3)^2 + 4 = (x^2 - 6x + 9) + 4 = x^2 - 6x + 13 $$ 7. **Write the polynomial as product of factors:** $$ m(x) = x (x - 5) (x^2 - 6x + 13) $$ 8. **Expand $x (x - 5)$:** $$ x (x - 5) = x^2 - 5x $$ 9. **Multiply $(x^2 - 5x)(x^2 - 6x + 13)$:** $$ (x^2 - 5x)(x^2 - 6x + 13) = x^2(x^2 - 6x + 13) - 5x(x^2 - 6x + 13) $$ $$ = x^4 - 6x^3 + 13x^2 - 5x^3 + 30x^2 - 65x $$ 10. **Combine like terms:** $$ x^4 - (6x^3 + 5x^3) + (13x^2 + 30x^2) - 65x = x^4 - 11x^3 + 43x^2 - 65x $$ 11. **Final polynomial:** $$ m(x) = x^4 - 11x^3 + 43x^2 - 65x $$ This is a possible polynomial function of degree 4 with the given zeros.