Divide using synthetic division $$ ( 8x^{2}+3x-11 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-4&8&3&-11\\& & -32& \color{black}{116} \\ \hline &\color{blue}{8}&\color{blue}{-29}&\color{orangered}{105} \end{array} $$The solution is:
$$ \frac{ 8x^{2}+3x-11 }{ x+4 } = \color{blue}{8x-29} ~+~ \frac{ \color{red}{ 105 } }{ x+4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-4}&8&3&-11\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-4&\color{orangered}{ 8 }&3&-11\\& & & \\ \hline &\color{orangered}{8}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 8 } = \color{blue}{ -32 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&8&3&-11\\& & \color{blue}{-32} & \\ \hline &\color{blue}{8}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -32 \right) } = \color{orangered}{ -29 } $
$$ \begin{array}{c|rrr}-4&8&\color{orangered}{ 3 }&-11\\& & \color{orangered}{-32} & \\ \hline &8&\color{orangered}{-29}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -29 \right) } = \color{blue}{ 116 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&8&3&-11\\& & -32& \color{blue}{116} \\ \hline &8&\color{blue}{-29}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -11 } + \color{orangered}{ 116 } = \color{orangered}{ 105 } $
$$ \begin{array}{c|rrr}-4&8&3&\color{orangered}{ -11 }\\& & -32& \color{orangered}{116} \\ \hline &\color{blue}{8}&\color{blue}{-29}&\color{orangered}{105} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x-29 } $ with a remainder of $ \color{red}{ 105 } $.