1. **State the problem:** We are given the polynomial $$f(x) = 4x^3 + 15x^2 - 63x - 54$$ and told that one zero is $$x = -6$$. We need to find another zero of $$f(x)$$. 2. **Use the Factor Theorem:** Since $$x = -6$$ is a zero, $$x + 6$$ is a factor of $$f(x)$$. We can divide $$f(x)$$ by $$x + 6$$ to find the quotient polynomial. 3. **Perform synthetic division:** Set up synthetic division with root $$-6$$: Coefficients: 4, 15, -63, -54 - Bring down 4. - Multiply 4 by -6: $$4 \times (-6) = -24$$. - Add to 15: $$15 + (-24) = -9$$. - Multiply -9 by -6: $$-9 \times (-6) = 54$$. - Add to -63: $$-63 + 54 = -9$$. - Multiply -9 by -6: $$-9 \times (-6) = 54$$. - Add to -54: $$-54 + 54 = 0$$ (remainder). The quotient polynomial is $$4x^2 - 9x - 9$$. 4. **Find zeros of the quotient:** Solve $$4x^2 - 9x - 9 = 0$$. Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-(-9) \pm \sqrt{(-9)^2 - 4 \times 4 \times (-9)}}{2 \times 4}$$ Calculate discriminant: $$81 + 144 = 225$$ So, $$x = \frac{9 \pm \sqrt{225}}{8} = \frac{9 \pm 15}{8}$$ 5. **Calculate the two roots:** - $$x = \frac{9 + 15}{8} = \frac{24}{8} = 3$$ - $$x = \frac{9 - 15}{8} = \frac{-6}{8} = -\frac{3}{4}$$ 6. **Answer:** Among the choices, $$3$$ is listed and is a zero of $$f(x)$$. **Final answer:** $$\boxed{3}$$