Divide using synthetic division $$ ( 8x^{4}-24x^{3}+3x^{2}-15x+11 ) \div ( x+3 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-3&8&-24&3&-15&11\\& & -24& 144& -441& \color{black}{1368} \\ \hline &\color{blue}{8}&\color{blue}{-48}&\color{blue}{147}&\color{blue}{-456}&\color{orangered}{1379} \end{array} $$The solution is:
$$ \dfrac{ 8x^{4}-24x^{3}+3x^{2}-15x+11 }{ x+3 } = \color{blue}{8x^{3}-48x^{2}+147x-456} ~+~ \dfrac{ \color{red}{ 1379 } }{ x+3 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&8&-24&3&-15&11\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 8 }&-24&3&-15&11\\& & & & & \\ \hline &\color{orangered}{8}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 8 } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&8&-24&3&-15&11\\& & \color{blue}{-24} & & & \\ \hline &\color{blue}{8}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -48 } $
$$ \begin{array}{c|rrrrr}-3&8&\color{orangered}{ -24 }&3&-15&11\\& & \color{orangered}{-24} & & & \\ \hline &8&\color{orangered}{-48}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -48 \right) } = \color{blue}{ 144 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&8&-24&3&-15&11\\& & -24& \color{blue}{144} & & \\ \hline &8&\color{blue}{-48}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 144 } = \color{orangered}{ 147 } $
$$ \begin{array}{c|rrrrr}-3&8&-24&\color{orangered}{ 3 }&-15&11\\& & -24& \color{orangered}{144} & & \\ \hline &8&-48&\color{orangered}{147}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 147 } = \color{blue}{ -441 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&8&-24&3&-15&11\\& & -24& 144& \color{blue}{-441} & \\ \hline &8&-48&\color{blue}{147}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ \left( -441 \right) } = \color{orangered}{ -456 } $
$$ \begin{array}{c|rrrrr}-3&8&-24&3&\color{orangered}{ -15 }&11\\& & -24& 144& \color{orangered}{-441} & \\ \hline &8&-48&147&\color{orangered}{-456}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -456 \right) } = \color{blue}{ 1368 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&8&-24&3&-15&11\\& & -24& 144& -441& \color{blue}{1368} \\ \hline &8&-48&147&\color{blue}{-456}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ 1368 } = \color{orangered}{ 1379 } $
$$ \begin{array}{c|rrrrr}-3&8&-24&3&-15&\color{orangered}{ 11 }\\& & -24& 144& -441& \color{orangered}{1368} \\ \hline &\color{blue}{8}&\color{blue}{-48}&\color{blue}{147}&\color{blue}{-456}&\color{orangered}{1379} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{3}-48x^{2}+147x-456 } $ with a remainder of $ \color{red}{ 1379 } $.