Question: Divide the following expression.
$$\frac{2x^3 - 6x^2 + 12x}{2x}$$
1. **State the problem:** We need to divide the polynomial expression $$2x^3 - 6x^2 + 12x$$ by $$2x$$.
2. **Write the division as a fraction:**
$$\frac{2x^3 - 6x^2 + 12x}{2x}$$
3. **Use the rule:** When dividing a polynomial by a monomial, divide each term in the numerator by the denominator separately.
4. **Divide each term:**
$$\frac{2x^3}{2x} - \frac{6x^2}{2x} + \frac{12x}{2x}$$
5. **Simplify each term:**
- $$\frac{2x^3}{2x} = \cancel{\frac{2}{2}} x^{3-1} = x^2$$
- $$\frac{6x^2}{2x} = \cancel{\frac{6}{2}} x^{2-1} = 3x$$
- $$\frac{12x}{2x} = \cancel{\frac{12}{2}} \cancel{\frac{x}{x}} = 6$$
6. **Combine the simplified terms:**
$$x^2 - 3x + 6$$
**Final answer:** $$x^2 - 3x + 6$$