a) \( m^{2}+5 m+9 \div m+2 \)

To divide the polynomial \( m^2 + 5m + 9 \) by \( m + 2 \), we can use polynomial long division. 1. **Set up the division**: Write \( m^2 + 5m + 9 \) under the long division symbol and \( m + 2 \) outside. 2. **Divide the leading term**: Divide the leading term of the dividend \( m^2 \) by the leading term of the divisor \( m \): \[ \frac{m^2}{m} = m \] 3. **Multiply and subtract**: Multiply \( m \) by the entire divisor \( m + 2 \): \[ m(m + 2) = m^2 + 2m \] Now subtract this from the original polynomial: \[ (m^2 + 5m + 9) - (m^2 + 2m) = (5m - 2m) + 9 = 3m + 9 \] 4. **Repeat the process**: Now divide the leading term of the new polynomial \( 3m \) by the leading term of the divisor \( m \): \[ \frac{3m}{m} = 3 \] Multiply \( 3 \) by the entire divisor \( m + 2 \): \[ 3(m + 2) = 3m + 6 \] Subtract this from \( 3m + 9 \): \[ (3m + 9) - (3m + 6) = 9 - 6 = 3 \] 5. **Final result**: Now we have a remainder of \( 3 \). Therefore, we can express the result of the division as: \[ m + 3 + \frac{3}{m + 2} \] So, the final answer is: \[ \frac{m^2 + 5m + 9}{m + 2} = m + 3 + \frac{3}{m + 2} \]