Divide using synthetic division $$ ( 6x^{4}-25x^{3}-25x^{2} ) \div ( x+5 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-5&6&-25&-25&0&0\\& & -30& 275& -1250& \color{black}{6250} \\ \hline &\color{blue}{6}&\color{blue}{-55}&\color{blue}{250}&\color{blue}{-1250}&\color{orangered}{6250} \end{array} $$The solution is:
$$ \dfrac{ 6x^{4}-25x^{3}-25x^{2} }{ x+5 } = \color{blue}{6x^{3}-55x^{2}+250x-1250} ~+~ \dfrac{ \color{red}{ 6250 } }{ x+5 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&6&-25&-25&0&0\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-5&\color{orangered}{ 6 }&-25&-25&0&0\\& & & & & \\ \hline &\color{orangered}{6}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 6 } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&6&-25&-25&0&0\\& & \color{blue}{-30} & & & \\ \hline &\color{blue}{6}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ -55 } $
$$ \begin{array}{c|rrrrr}-5&6&\color{orangered}{ -25 }&-25&0&0\\& & \color{orangered}{-30} & & & \\ \hline &6&\color{orangered}{-55}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -55 \right) } = \color{blue}{ 275 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&6&-25&-25&0&0\\& & -30& \color{blue}{275} & & \\ \hline &6&\color{blue}{-55}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ 275 } = \color{orangered}{ 250 } $
$$ \begin{array}{c|rrrrr}-5&6&-25&\color{orangered}{ -25 }&0&0\\& & -30& \color{orangered}{275} & & \\ \hline &6&-55&\color{orangered}{250}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 250 } = \color{blue}{ -1250 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&6&-25&-25&0&0\\& & -30& 275& \color{blue}{-1250} & \\ \hline &6&-55&\color{blue}{250}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -1250 \right) } = \color{orangered}{ -1250 } $
$$ \begin{array}{c|rrrrr}-5&6&-25&-25&\color{orangered}{ 0 }&0\\& & -30& 275& \color{orangered}{-1250} & \\ \hline &6&-55&250&\color{orangered}{-1250}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -1250 \right) } = \color{blue}{ 6250 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&6&-25&-25&0&0\\& & -30& 275& -1250& \color{blue}{6250} \\ \hline &6&-55&250&\color{blue}{-1250}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 6250 } = \color{orangered}{ 6250 } $
$$ \begin{array}{c|rrrrr}-5&6&-25&-25&0&\color{orangered}{ 0 }\\& & -30& 275& -1250& \color{orangered}{6250} \\ \hline &\color{blue}{6}&\color{blue}{-55}&\color{blue}{250}&\color{blue}{-1250}&\color{orangered}{6250} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{3}-55x^{2}+250x-1250 } $ with a remainder of $ \color{red}{ 6250 } $.