Divide using synthetic division $$ ( x^{4}-12x^{3}+43x^{2}-112x+192 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-3&1&-12&43&-112&192\\& & -3& 45& -264& \color{black}{1128} \\ \hline &\color{blue}{1}&\color{blue}{-15}&\color{blue}{88}&\color{blue}{-376}&\color{orangered}{1320} \end{array} $$The solution is:
$$ \frac{ x^{4}-12x^{3}+43x^{2}-112x+192 }{ x+3 } = \color{blue}{x^{3}-15x^{2}+88x-376} ~+~ \frac{ \color{red}{ 1320 } }{ x+3 } $$Explanation
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.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-12&43&-112&192\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 1 }&-12&43&-112&192\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 1 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-12&43&-112&192\\& & \color{blue}{-3} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -15 } $
$$ \begin{array}{c|rrrrr}-3&1&\color{orangered}{ -12 }&43&-112&192\\& & \color{orangered}{-3} & & & \\ \hline &1&\color{orangered}{-15}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -15 \right) } = \color{blue}{ 45 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-12&43&-112&192\\& & -3& \color{blue}{45} & & \\ \hline &1&\color{blue}{-15}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 43 } + \color{orangered}{ 45 } = \color{orangered}{ 88 } $
$$ \begin{array}{c|rrrrr}-3&1&-12&\color{orangered}{ 43 }&-112&192\\& & -3& \color{orangered}{45} & & \\ \hline &1&-15&\color{orangered}{88}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 88 } = \color{blue}{ -264 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-12&43&-112&192\\& & -3& 45& \color{blue}{-264} & \\ \hline &1&-15&\color{blue}{88}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -112 } + \color{orangered}{ \left( -264 \right) } = \color{orangered}{ -376 } $
$$ \begin{array}{c|rrrrr}-3&1&-12&43&\color{orangered}{ -112 }&192\\& & -3& 45& \color{orangered}{-264} & \\ \hline &1&-15&88&\color{orangered}{-376}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -376 \right) } = \color{blue}{ 1128 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-12&43&-112&192\\& & -3& 45& -264& \color{blue}{1128} \\ \hline &1&-15&88&\color{blue}{-376}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 192 } + \color{orangered}{ 1128 } = \color{orangered}{ 1320 } $
$$ \begin{array}{c|rrrrr}-3&1&-12&43&-112&\color{orangered}{ 192 }\\& & -3& 45& -264& \color{orangered}{1128} \\ \hline &\color{blue}{1}&\color{blue}{-15}&\color{blue}{88}&\color{blue}{-376}&\color{orangered}{1320} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-15x^{2}+88x-376 } $ with a remainder of $ \color{red}{ 1320 } $.