Solve. \( 4 m^{3}+4 m^{2}-8 m=0 \)

To solve the equation: \[ 4m^{3} + 4m^{2} - 8m = 0 \] **Step 1: Factor out the Greatest Common Factor (GCF)** First, identify and factor out the GCF from all terms. In this case, the GCF is \(4m\): \[ 4m(m^{2} + m - 2) = 0 \] **Step 2: Apply the Zero Product Property** The Zero Product Property states that if the product of multiple factors is zero, at least one of the factors must be zero. Therefore, set each factor equal to zero: 1. \(4m = 0\) 2. \(m^{2} + m - 2 = 0\) **Solution for the First Factor:** \[ 4m = 0 \\ m = 0 \] **Solution for the Quadratic Equation:** \[ m^{2} + m - 2 = 0 \] To solve the quadratic equation \(m^{2} + m - 2 = 0\), you can use the quadratic formula: \[ m = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] where \(a = 1\), \(b = 1\), and \(c = -2\). Plugging in the values: \[ m = \frac{-1 \pm \sqrt{(1)^2 - 4(1)(-2)}}{2(1)} \\ m = \frac{-1 \pm \sqrt{1 + 8}}{2} \\ m = \frac{-1 \pm \sqrt{9}}{2} \\ m = \frac{-1 \pm 3}{2} \] This gives two solutions: 1. \(m = \frac{-1 + 3}{2} = \frac{2}{2} = 1\) 2. \(m = \frac{-1 - 3}{2} = \frac{-4}{2} = -2\) **Final Solutions:** \[ m = 0, \quad m = 1, \quad \text{and} \quad m = -2 \] **Verification:** It's always good practice to verify the solutions by substituting them back into the original equation. 1. **For \(m = 0\):** \[ 4(0)^3 + 4(0)^2 - 8(0) = 0 + 0 - 0 = 0 \] 2. **For \(m = 1\):** \[ 4(1)^3 + 4(1)^2 - 8(1) = 4 + 4 - 8 = 0 \] 3. **For \(m = -2\):** \[ 4(-2)^3 + 4(-2)^2 - 8(-2) = 4(-8) + 4(4) + 16 = -32 + 16 + 16 = 0 \] All solutions satisfy the original equation. **Conclusion:** The solutions to the equation \(4m^{3} + 4m^{2} - 8m = 0\) are: \[ m = 0, \quad m = 1, \quad \text{and} \quad m = -2 \]