Divide using synthetic division $$ ( 2x^{4}+13x^{3}+14x^{2}-57 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-3&2&13&14&0&-57\\& & -6& -21& 21& \color{black}{-63} \\ \hline &\color{blue}{2}&\color{blue}{7}&\color{blue}{-7}&\color{blue}{21}&\color{orangered}{-120} \end{array} $$The solution is:
$$ \frac{ 2x^{4}+13x^{3}+14x^{2}-57 }{ x+3 } = \color{blue}{2x^{3}+7x^{2}-7x+21} \color{red}{~-~} \frac{ \color{red}{ 120 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&2&13&14&0&-57\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 2 }&13&14&0&-57\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 2 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&2&13&14&0&-57\\& & \color{blue}{-6} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 13 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrrr}-3&2&\color{orangered}{ 13 }&14&0&-57\\& & \color{orangered}{-6} & & & \\ \hline &2&\color{orangered}{7}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 7 } = \color{blue}{ -21 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&2&13&14&0&-57\\& & -6& \color{blue}{-21} & & \\ \hline &2&\color{blue}{7}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ \left( -21 \right) } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrrr}-3&2&13&\color{orangered}{ 14 }&0&-57\\& & -6& \color{orangered}{-21} & & \\ \hline &2&7&\color{orangered}{-7}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ 21 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&2&13&14&0&-57\\& & -6& -21& \color{blue}{21} & \\ \hline &2&7&\color{blue}{-7}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 21 } = \color{orangered}{ 21 } $
$$ \begin{array}{c|rrrrr}-3&2&13&14&\color{orangered}{ 0 }&-57\\& & -6& -21& \color{orangered}{21} & \\ \hline &2&7&-7&\color{orangered}{21}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 21 } = \color{blue}{ -63 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&2&13&14&0&-57\\& & -6& -21& 21& \color{blue}{-63} \\ \hline &2&7&-7&\color{blue}{21}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -57 } + \color{orangered}{ \left( -63 \right) } = \color{orangered}{ -120 } $
$$ \begin{array}{c|rrrrr}-3&2&13&14&0&\color{orangered}{ -57 }\\& & -6& -21& 21& \color{orangered}{-63} \\ \hline &\color{blue}{2}&\color{blue}{7}&\color{blue}{-7}&\color{blue}{21}&\color{orangered}{-120} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}+7x^{2}-7x+21 } $ with a remainder of $ \color{red}{ -120 } $.