Solve. $$ 4 m^{3}+4 m^{2}-8 m=0 $$

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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:

$$
m = 0, \quad m = 1, \quad \text{and} \quad m = -2
$$
The solutions are $$ m = 0 $$, $$ m = 1 $$, and $$ m = -2 $$.

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

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