Divide using synthetic division $$ ( 3x^{4}+250x ) \div ( x+6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-6&3&0&0&250&0\\& & -18& 108& -648& \color{black}{2388} \\ \hline &\color{blue}{3}&\color{blue}{-18}&\color{blue}{108}&\color{blue}{-398}&\color{orangered}{2388} \end{array} $$The solution is:
$$ \frac{ 3x^{4}+250x }{ x+6 } = \color{blue}{3x^{3}-18x^{2}+108x-398} ~+~ \frac{ \color{red}{ 2388 } }{ x+6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 6 = 0 $ ( $ x = \color{blue}{ -6 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&3&0&0&250&0\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-6&\color{orangered}{ 3 }&0&0&250&0\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 3 } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&3&0&0&250&0\\& & \color{blue}{-18} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -18 } $
$$ \begin{array}{c|rrrrr}-6&3&\color{orangered}{ 0 }&0&250&0\\& & \color{orangered}{-18} & & & \\ \hline &3&\color{orangered}{-18}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -18 \right) } = \color{blue}{ 108 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&3&0&0&250&0\\& & -18& \color{blue}{108} & & \\ \hline &3&\color{blue}{-18}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 108 } = \color{orangered}{ 108 } $
$$ \begin{array}{c|rrrrr}-6&3&0&\color{orangered}{ 0 }&250&0\\& & -18& \color{orangered}{108} & & \\ \hline &3&-18&\color{orangered}{108}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 108 } = \color{blue}{ -648 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&3&0&0&250&0\\& & -18& 108& \color{blue}{-648} & \\ \hline &3&-18&\color{blue}{108}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 250 } + \color{orangered}{ \left( -648 \right) } = \color{orangered}{ -398 } $
$$ \begin{array}{c|rrrrr}-6&3&0&0&\color{orangered}{ 250 }&0\\& & -18& 108& \color{orangered}{-648} & \\ \hline &3&-18&108&\color{orangered}{-398}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -398 \right) } = \color{blue}{ 2388 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&3&0&0&250&0\\& & -18& 108& -648& \color{blue}{2388} \\ \hline &3&-18&108&\color{blue}{-398}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 2388 } = \color{orangered}{ 2388 } $
$$ \begin{array}{c|rrrrr}-6&3&0&0&250&\color{orangered}{ 0 }\\& & -18& 108& -648& \color{orangered}{2388} \\ \hline &\color{blue}{3}&\color{blue}{-18}&\color{blue}{108}&\color{blue}{-398}&\color{orangered}{2388} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}-18x^{2}+108x-398 } $ with a remainder of $ \color{red}{ 2388 } $.