Question: Divide the following expression.
$$\frac{6x^3 + 8x^2 - 10x}{2x}$$
Options:
1. $$3x^2 + 4x - 5$$
2. $$x^2 + 4x - 10$$
3. $$3x^2 + 4x - 10$$
4. $$x^2 + 4x - 5$$
1. **State the problem:** We need to divide the polynomial $$6x^3 + 8x^2 - 10x$$ by $$2x$$.
2. **Write the division as a fraction:**
$$\frac{6x^3 + 8x^2 - 10x}{2x}$$
3. **Divide each term in the numerator by the denominator:**
$$\frac{6x^3}{2x} + \frac{8x^2}{2x} - \frac{10x}{2x}$$
4. **Simplify each term:**
- For $$\frac{6x^3}{2x}$$, divide coefficients and subtract exponents:
$$\frac{6}{2} x^{3-1} = 3x^2$$
- For $$\frac{8x^2}{2x}$$:
$$\frac{8}{2} x^{2-1} = 4x$$
- For $$\frac{10x}{2x}$$:
$$\frac{10}{2} x^{1-1} = 5$$
5. **Combine the simplified terms:**
$$3x^2 + 4x - 5$$
6. **Answer:** The quotient is $$3x^2 + 4x - 5$$.
This matches option 1.