\( 16 c^{4}-62 c^{3}+48 c^{2}-18 c \) enter \( 2 c 2+6 c+5 \)

To simplify the expression \( 16c^{4} - 62c^{3} + 48c^{2} - 18c \) by dividing it by \( 2c^2 + 6c + 5 \), we can perform polynomial long division. 1. **Set up the division**: We want to divide \( 16c^{4} - 62c^{3} + 48c^{2} - 18c \) by \( 2c^2 + 6c + 5 \). 2. **Divide the leading terms**: - The leading term of the dividend is \( 16c^4 \) and the leading term of the divisor is \( 2c^2 \). - Divide \( 16c^4 \) by \( 2c^2 \) to get \( 8c^2 \). 3. **Multiply and subtract**: - Multiply \( 8c^2 \) by \( 2c^2 + 6c + 5 \): \[ 8c^2(2c^2 + 6c + 5) = 16c^4 + 48c^3 + 40c^2 \] - Subtract this from the original polynomial: \[ (16c^{4} - 62c^{3} + 48c^{2} - 18c) - (16c^{4} + 48c^{3} + 40c^{2}) = -110c^{3} + 8c^{2} - 18c \] 4. **Repeat the process**: - Now divide the leading term \( -110c^3 \) by \( 2c^2 \) to get \( -55c \). - Multiply \( -55c \) by \( 2c^2 + 6c + 5 \): \[ -55c(2c^2 + 6c + 5) = -110c^3 - 330c^2 - 275c \] - Subtract this from the current polynomial: \[ (-110c^{3} + 8c^{2} - 18c) - (-110c^{3} - 330c^{2} - 275c) = 338c^{2} + 257c \] 5. **Continue the process**: - Divide \( 338c^2 \) by \( 2c^2 \) to get \( 169 \). - Multiply \( 169 \) by \( 2c^2 + 6c + 5 \): \[ 169(2c^2 + 6c + 5) = 338c^2 + 1014c + 845 \] - Subtract this from the current polynomial: \[ (338c^{2} + 257c) - (338c^{2} + 1014c + 845) = -757c - 845 \] 6. **Final result**: - The remainder is \( -757c - 845 \). - Thus, the result of the division is: \[ 8c^2 - 55c + 169 + \frac{-757c - 845}{2c^2 + 6c + 5} \] So, the final answer is: \[ 8c^2 - 55c + 169 - \frac{757c + 845}{2c^2 + 6c + 5} \]