Divide. \[ \left(15 x^{3}+7 x^{2}+18 x-13\right) \div(5 x-1) \] Your answer should give the quotient and the remainder.

To divide \( (15 x^{3}+7 x^{2}+18 x-13) \) by \( (5 x-1) \), we can use polynomial long division. 1. **Divide the leading term**: \( \frac{15 x^3}{5 x} = 3 x^2 \). 2. **Multiply and subtract**: \[ (5x - 1)(3x^2) = 15x^3 - 3x^2. \] Subtracting this from the original polynomial: \[ (15x^3 + 7x^2 + 18x - 13) - (15x^3 - 3x^2) = 10x^2 + 18x - 13. \] 3. **Repeat the process**: \[ \frac{10x^2}{5x} = 2x. \] Multiply and subtract: \[ (5x - 1)(2x) = 10x^2 - 2x. \] Subtract: \[ (10x^2 + 18x - 13) - (10x^2 - 2x) = 20x - 13. \] 4. **Repeat again**: \[ \frac{20x}{5x} = 4. \] Multiply and subtract: \[ (5x - 1)(4) = 20x - 4. \] Subtract: \[ (20x - 13) - (20x - 4) = -9. \] At this point, we can no longer divide as the degree of the remainder (-9) is less than the degree of the divisor (5x - 1). Thus, the quotient is \( 3x^2 + 2x + 4 \) and the remainder is \( -9 \). We can write the result as: \[ \boxed{3x^2 + 2x + 4 \quad \text{remainder} \quad -9.} \]