Divide using synthetic division $$ ( 6x^{3}-65x^{2}+206x-130 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-5&6&-65&206&-130\\& & -30& 475& \color{black}{-3405} \\ \hline &\color{blue}{6}&\color{blue}{-95}&\color{blue}{681}&\color{orangered}{-3535} \end{array} $$The solution is:
$$ \frac{ 6x^{3}-65x^{2}+206x-130 }{ x+5 } = \color{blue}{6x^{2}-95x+681} \color{red}{~-~} \frac{ \color{red}{ 3535 } }{ x+5 } $$Explanation
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.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&6&-65&206&-130\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-5&\color{orangered}{ 6 }&-65&206&-130\\& & & & \\ \hline &\color{orangered}{6}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&6&-65&206&-130\\& & \color{blue}{-30} & & \\ \hline &\color{blue}{6}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -65 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ -95 } $
$$ \begin{array}{c|rrrr}-5&6&\color{orangered}{ -65 }&206&-130\\& & \color{orangered}{-30} & & \\ \hline &6&\color{orangered}{-95}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -95 \right) } = \color{blue}{ 475 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&6&-65&206&-130\\& & -30& \color{blue}{475} & \\ \hline &6&\color{blue}{-95}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 206 } + \color{orangered}{ 475 } = \color{orangered}{ 681 } $
$$ \begin{array}{c|rrrr}-5&6&-65&\color{orangered}{ 206 }&-130\\& & -30& \color{orangered}{475} & \\ \hline &6&-95&\color{orangered}{681}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 681 } = \color{blue}{ -3405 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&6&-65&206&-130\\& & -30& 475& \color{blue}{-3405} \\ \hline &6&-95&\color{blue}{681}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -130 } + \color{orangered}{ \left( -3405 \right) } = \color{orangered}{ -3535 } $
$$ \begin{array}{c|rrrr}-5&6&-65&206&\color{orangered}{ -130 }\\& & -30& 475& \color{orangered}{-3405} \\ \hline &\color{blue}{6}&\color{blue}{-95}&\color{blue}{681}&\color{orangered}{-3535} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{2}-95x+681 } $ with a remainder of $ \color{red}{ -3535 } $.