Use long division to divide. \[ \frac{15 x^{4}+5 x^{3}+16 x^{2}+5 x+8}{3 x^{2}+x+2}=\square \]

To divide the polynomial \( \frac{15x^4 + 5x^3 + 16x^2 + 5x + 8}{3x^2 + x + 2} \) using long division, we follow these steps: 1. **Divide the leading term** of the dividend \( 15x^4 \) by the leading term of the divisor \( 3x^2 \). This gives us \( 5x^2 \). 2. **Multiply** \( 5x^2 \) by the entire divisor \( (3x^2 + x + 2) \): \[ 5x^2(3x^2 + x + 2) = 15x^4 + 5x^3 + 10x^2 \] 3. **Subtract** this result from the original polynomial: \[ (15x^4 + 5x^3 + 16x^2 + 5x + 8) - (15x^4 + 5x^3 + 10x^2) = (16x^2 - 10x^2) + 5x + 8 = 6x^2 + 5x + 8 \] 4. **Repeat the process**: Now, divide the leading term \( 6x^2 \) by \( 3x^2 \) to get \( 2 \). 5. **Multiply** \( 2 \) by \( (3x^2 + x + 2) \): \[ 2(3x^2 + x + 2) = 6x^2 + 2x + 4 \] 6. **Subtract** this from the current polynomial: \[ (6x^2 + 5x + 8) - (6x^2 + 2x + 4) = (5x - 2x) + (8 - 4) = 3x + 4 \] The remainder is \( 3x + 4 \), which is of lower degree than the divisor \( 3x^2 + x + 2 \). So, the result of the division is: \[ 5x^2 + 2 + \frac{3x + 4}{3x^2 + x + 2} \] Thus, we can write: \[ \frac{15 x^{4}+5 x^{3}+16 x^{2}+5 x+8}{3 x^{2}+x+2} = 5x^2 + 2 + \frac{3x + 4}{3x^2 + x + 2} \]