Use the long division method to find the result when \( 8 x^{3}+16 x^{2}+16 x+15 \) is divided by \( 2 x+3 \). If there is a remainder, express the result in the form \( q(x)+\frac{r(x)}{b(x)} \).

To perform the long division of \( 8x^3 + 16x^2 + 16x + 15 \) by \( 2x + 3 \), follow these steps: 1. **Set up the long division**: Write \( 8x^3 + 16x^2 + 16x + 15 \) under the long division symbol and \( 2x + 3 \) outside. 2. **Divide the leading terms**: Divide the leading term of the dividend \( 8x^3 \) by the leading term of the divisor \( 2x \) to get \( 4x^2 \). 3. **Multiply**: Multiply \( 4x^2 \) by \( 2x + 3 \) to get \( 8x^3 + 12x^2 \). 4. **Subtract**: Subtract \( (8x^3 + 12x^2) \) from \( (8x^3 + 16x^2 + 16x + 15) \): \[ (8x^3 + 16x^2 + 16x + 15) - (8x^3 + 12x^2) = 4x^2 + 16x + 15. \] 5. **Repeat the process**: Divide the new leading term \( 4x^2 \) by \( 2x \) to get \( 2x \). Multiply \( 2x \) by \( 2x + 3 \) to get \( 4x^2 + 6x \). Subtract again: \[ (4x^2 + 16x + 15) - (4x^2 + 6x) = 10x + 15. \] 6. **Continue dividing**: Now divide \( 10x \) by \( 2x \) to get \( 5 \). Multiply \( 5 \) by \( 2x + 3 \) to get \( 10x + 15 \). Subtract: \[ (10x + 15) - (10x + 15) = 0. \] Since we have a remainder of 0, the result of the division is: \[ q(x) = 4x^2 + 2x + 5. \] Thus, the final answer is: \[ 4x^2 + 2x + 5 + \frac{0}{2x+3} = 4x^2 + 2x + 5. \]