Use long division to divide. Express any numbers as integers or simplified fractions \[ \left(7 x^{3}-2 x^{2}+5\right) \div(2 x+1)=\square \]

To divide \( (7x^3 - 2x^2 + 5) \) by \( (2x + 1) \) using long division: 1. Divide the first term of the dividend \( 7x^3 \) by the first term of the divisor \( 2x \) to get \( \frac{7}{2}x^2 \). 2. Multiply \( \frac{7}{2}x^2 \) by \( (2x + 1) \) to get \( 7x^3 + \frac{7}{2}x^2 \). 3. Subtract this from the original polynomial: \[ (7x^3 - 2x^2 + 5) - (7x^3 + \frac{7}{2}x^2) = -2x^2 - \frac{7}{2}x^2 + 5 = -\frac{11}{2}x^2 + 5. \] 4. Bring down the next term, which is already included, so we just rewrite it. 5. Next, divide \( -\frac{11}{2}x^2 \) by \( 2x \) to get \( -\frac{11}{4}x \). 6. Multiply \( -\frac{11}{4}x \) by \( (2x + 1) \) to get \( -\frac{11}{2}x^2 - \frac{11}{4}x \). 7. Subtract: \[ \left(-\frac{11}{2}x^2 + 5\right) - \left(-\frac{11}{2}x^2 - \frac{11}{4}x\right) = \frac{11}{4}x + 5. \] 8. Divide \( \frac{11}{4}x \) by \( 2x \) to get \( \frac{11}{8} \). 9. Multiply \( \frac{11}{8} \) by \( (2x + 1) \) to get \( \frac{11}{4}x + \frac{11}{8} \). 10. Finally, subtract: \[ \left(\frac{11}{4}x + 5\right) - \left(\frac{11}{4}x + \frac{11}{8}\right) = 5 - \frac{11}{8} = \frac{40}{8} - \frac{11}{8} = \frac{29}{8}. \] So, the remainder is \( \frac{29}{8} \). Thus, the full answer is: \[ \left(7x^{3}-2x^{2}+5\right) \div(2x+1) = \frac{7}{2}x^2 - \frac{11}{4}x + \frac{11}{8} + \frac{\frac{29}{8}}{2x + 1}. \]