Divide using synthetic division $$ ( x^{2}-49x+29 ) \div ( 4x+12 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-12&1&-49&29\\& & -12& \color{black}{732} \\ \hline &\color{blue}{1}&\color{blue}{-61}&\color{orangered}{761} \end{array} $$The solution is:
$$ \frac{ x^{2}-49x+29 }{ x+12 } = \color{blue}{x-61} ~+~ \frac{ \color{red}{ 761 } }{ x+12 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 12 = 0 $ ( $ x = \color{blue}{ -12 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-12}&1&-49&29\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-12&\color{orangered}{ 1 }&-49&29\\& & & \\ \hline &\color{orangered}{1}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -12 } \cdot \color{blue}{ 1 } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrr}\color{blue}{-12}&1&-49&29\\& & \color{blue}{-12} & \\ \hline &\color{blue}{1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -49 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -61 } $
$$ \begin{array}{c|rrr}-12&1&\color{orangered}{ -49 }&29\\& & \color{orangered}{-12} & \\ \hline &1&\color{orangered}{-61}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -12 } \cdot \color{blue}{ \left( -61 \right) } = \color{blue}{ 732 } $.
$$ \begin{array}{c|rrr}\color{blue}{-12}&1&-49&29\\& & -12& \color{blue}{732} \\ \hline &1&\color{blue}{-61}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 29 } + \color{orangered}{ 732 } = \color{orangered}{ 761 } $
$$ \begin{array}{c|rrr}-12&1&-49&\color{orangered}{ 29 }\\& & -12& \color{orangered}{732} \\ \hline &\color{blue}{1}&\color{blue}{-61}&\color{orangered}{761} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x-61 } $ with a remainder of $ \color{red}{ 761 } $.