Find remainder when $ 4x^{2}+2x+5 $ is divided by $ -5x+6 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-6&4&2&5\\& & -24& \color{black}{132} \\ \hline &\color{blue}{4}&\color{blue}{-22}&\color{orangered}{137} \end{array} $$The remainder when $ 4x^{2}+2x+5 $ is divided by $ x+6 $ is $ \, \color{red}{ 137 } $.
Explanation
We can find remainder using synthetic division method.
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 6 = 0 $ ( $ x = \color{blue}{ -6 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-6}&4&2&5\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-6&\color{orangered}{ 4 }&2&5\\& & & \\ \hline &\color{orangered}{4}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 4 } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrr}\color{blue}{-6}&4&2&5\\& & \color{blue}{-24} & \\ \hline &\color{blue}{4}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -22 } $
$$ \begin{array}{c|rrr}-6&4&\color{orangered}{ 2 }&5\\& & \color{orangered}{-24} & \\ \hline &4&\color{orangered}{-22}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -22 \right) } = \color{blue}{ 132 } $.
$$ \begin{array}{c|rrr}\color{blue}{-6}&4&2&5\\& & -24& \color{blue}{132} \\ \hline &4&\color{blue}{-22}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 132 } = \color{orangered}{ 137 } $
$$ \begin{array}{c|rrr}-6&4&2&\color{orangered}{ 5 }\\& & -24& \color{orangered}{132} \\ \hline &\color{blue}{4}&\color{blue}{-22}&\color{orangered}{137} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 137 }\right) $.