Divide using synthetic division $$ ( 2x^{3}+x^{2}-38x+35 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-5&2&1&-38&35\\& & -10& 45& \color{black}{-35} \\ \hline &\color{blue}{2}&\color{blue}{-9}&\color{blue}{7}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 2x^{3}+x^{2}-38x+35 }{ x+5 } = \color{blue}{2x^{2}-9x+7} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&2&1&-38&35\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-5&\color{orangered}{ 2 }&1&-38&35\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 2 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&2&1&-38&35\\& & \color{blue}{-10} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrr}-5&2&\color{orangered}{ 1 }&-38&35\\& & \color{orangered}{-10} & & \\ \hline &2&\color{orangered}{-9}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 45 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&2&1&-38&35\\& & -10& \color{blue}{45} & \\ \hline &2&\color{blue}{-9}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -38 } + \color{orangered}{ 45 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrr}-5&2&1&\color{orangered}{ -38 }&35\\& & -10& \color{orangered}{45} & \\ \hline &2&-9&\color{orangered}{7}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 7 } = \color{blue}{ -35 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&2&1&-38&35\\& & -10& 45& \color{blue}{-35} \\ \hline &2&-9&\color{blue}{7}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 35 } + \color{orangered}{ \left( -35 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-5&2&1&-38&\color{orangered}{ 35 }\\& & -10& 45& \color{orangered}{-35} \\ \hline &\color{blue}{2}&\color{blue}{-9}&\color{blue}{7}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}-9x+7 } $ with a remainder of $ \color{red}{ 0 } $.