1. **State the problem:** We are given that $(2x + 3)$ is a factor of the cubic polynomial $4x^3 - 12x^2 - 7x + 30$. We need to find all three solutions to the equation $4x^3 - 12x^2 - 7x + 30 = 0$. 2. **Use the factor to find one root:** Since $2x + 3 = 0$ is a factor, solve for $x$: $$2x + 3 = 0 \implies x = -\frac{3}{2}$$ This is one root of the cubic equation. 3. **Divide the cubic polynomial by $2x + 3$ to find the quadratic factor:** Use polynomial division or synthetic division to divide $4x^3 - 12x^2 - 7x + 30$ by $2x + 3$. Set up the division: Divide $4x^3$ by $2x$ to get $2x^2$. Multiply $2x^2$ by $2x + 3$: $$2x^2 \times (2x + 3) = 4x^3 + 6x^2$$ Subtract this from the original polynomial: $$\cancel{4x^3} - 12x^2 - 7x + 30 - (\cancel{4x^3} + 6x^2) = -18x^2 - 7x + 30$$ Next, divide $-18x^2$ by $2x$ to get $-9x$. Multiply $-9x$ by $2x + 3$: $$-9x \times (2x + 3) = -18x^2 - 27x$$ Subtract: $$-18x^2 - 7x + 30 - (-18x^2 - 27x) = 20x + 30$$ Divide $20x$ by $2x$ to get $10$. Multiply $10$ by $2x + 3$: $$10 \times (2x + 3) = 20x + 30$$ Subtract: $$20x + 30 - (20x + 30) = 0$$ So the quotient is $2x^2 - 9x + 10$. 4. **Solve the quadratic equation:** $$2x^2 - 9x + 10 = 0$$ Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=2$, $b=-9$, $c=10$. Calculate the discriminant: $$\Delta = (-9)^2 - 4 \times 2 \times 10 = 81 - 80 = 1$$ Calculate the roots: $$x = \frac{9 \pm \sqrt{1}}{2 \times 2} = \frac{9 \pm 1}{4}$$ So, $$x_1 = \frac{9 + 1}{4} = \frac{10}{4} = \frac{5}{2}$$ $$x_2 = \frac{9 - 1}{4} = \frac{8}{4} = 2$$ 5. **Final solutions:** The three solutions to $4x^3 - 12x^2 - 7x + 30 = 0$ are: $$x = -\frac{3}{2}, \quad x = 2, \quad x = \frac{5}{2}$$