Divide using synthetic division $$ ( 2x^{3}+6x^{2}+2x-10 ) \div ( x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}1&2&6&2&-10\\& & 2& 8& \color{black}{10} \\ \hline &\color{blue}{2}&\color{blue}{8}&\color{blue}{10}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 2x^{3}+6x^{2}+2x-10 }{ x-1 } = \color{blue}{2x^{2}+8x+10} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -1 = 0 $ ( $ x = \color{blue}{ 1 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{1}&2&6&2&-10\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}1&\color{orangered}{ 2 }&6&2&-10\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 2 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&2&6&2&-10\\& & \color{blue}{2} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 2 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrr}1&2&\color{orangered}{ 6 }&2&-10\\& & \color{orangered}{2} & & \\ \hline &2&\color{orangered}{8}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 8 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&2&6&2&-10\\& & 2& \color{blue}{8} & \\ \hline &2&\color{blue}{8}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 8 } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrr}1&2&6&\color{orangered}{ 2 }&-10\\& & 2& \color{orangered}{8} & \\ \hline &2&8&\color{orangered}{10}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 10 } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{1}&2&6&2&-10\\& & 2& 8& \color{blue}{10} \\ \hline &2&8&\color{blue}{10}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 10 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}1&2&6&2&\color{orangered}{ -10 }\\& & 2& 8& \color{orangered}{10} \\ \hline &\color{blue}{2}&\color{blue}{8}&\color{blue}{10}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}+8x+10 } $ with a remainder of $ \color{red}{ 0 } $.