Divide using synthetic division $$ ( -4x^{3}+x^{2}+13x ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-3&-4&1&13&0\\& & 12& -39& \color{black}{78} \\ \hline &\color{blue}{-4}&\color{blue}{13}&\color{blue}{-26}&\color{orangered}{78} \end{array} $$The solution is:
$$ \frac{ -4x^{3}+x^{2}+13x }{ x+3 } = \color{blue}{-4x^{2}+13x-26} ~+~ \frac{ \color{red}{ 78 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&-4&1&13&0\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-3&\color{orangered}{ -4 }&1&13&0\\& & & & \\ \hline &\color{orangered}{-4}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&-4&1&13&0\\& & \color{blue}{12} & & \\ \hline &\color{blue}{-4}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 12 } = \color{orangered}{ 13 } $
$$ \begin{array}{c|rrrr}-3&-4&\color{orangered}{ 1 }&13&0\\& & \color{orangered}{12} & & \\ \hline &-4&\color{orangered}{13}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 13 } = \color{blue}{ -39 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&-4&1&13&0\\& & 12& \color{blue}{-39} & \\ \hline &-4&\color{blue}{13}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 13 } + \color{orangered}{ \left( -39 \right) } = \color{orangered}{ -26 } $
$$ \begin{array}{c|rrrr}-3&-4&1&\color{orangered}{ 13 }&0\\& & 12& \color{orangered}{-39} & \\ \hline &-4&13&\color{orangered}{-26}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -26 \right) } = \color{blue}{ 78 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&-4&1&13&0\\& & 12& -39& \color{blue}{78} \\ \hline &-4&13&\color{blue}{-26}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 78 } = \color{orangered}{ 78 } $
$$ \begin{array}{c|rrrr}-3&-4&1&13&\color{orangered}{ 0 }\\& & 12& -39& \color{orangered}{78} \\ \hline &\color{blue}{-4}&\color{blue}{13}&\color{blue}{-26}&\color{orangered}{78} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -4x^{2}+13x-26 } $ with a remainder of $ \color{red}{ 78 } $.