1. **State the problem:** We need to find all x-intercepts of the function $$f(x) = -4x^4 + 2x^3 + 12x^2$$. The x-intercepts occur where $$f(x) = 0$$. 2. **Set the function equal to zero:** $$-4x^4 + 2x^3 + 12x^2 = 0$$ 3. **Factor out the greatest common factor (GCF):** The GCF of the terms is $$2x^2$$, so factor it out: $$2x^2(-2x^2 + x + 6) = 0$$ 4. **Set each factor equal to zero:** - For $$2x^2 = 0$$, divide both sides by 2: $$x^2 = 0$$ So, $$x = 0$$. - For $$-2x^2 + x + 6 = 0$$, multiply both sides by $$-1$$ to simplify: $$2x^2 - x - 6 = 0$$ 5. **Solve the quadratic equation $$2x^2 - x - 6 = 0$$ using the quadratic formula:** The quadratic formula is: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=2$$, $$b=-1$$, and $$c=-6$$. Calculate the discriminant: $$b^2 - 4ac = (-1)^2 - 4(2)(-6) = 1 + 48 = 49$$ Calculate the roots: $$x = \frac{-(-1) \pm \sqrt{49}}{2 \times 2} = \frac{1 \pm 7}{4}$$ So, - $$x = \frac{1 + 7}{4} = \frac{8}{4} = 2$$ - $$x = \frac{1 - 7}{4} = \frac{-6}{4} = -\frac{3}{2}$$ 6. **List all x-intercepts:** $$x = 0, 2, -\frac{3}{2}$$ **Final answer:** The x-intercepts of the function are $$x = 0$$, $$x = 2$$, and $$x = -\frac{3}{2}$$.