Divide using synthetic division $$ ( 2x^{3}-5x^{2}+x+1 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-4&2&-5&1&1\\& & -8& 52& \color{black}{-212} \\ \hline &\color{blue}{2}&\color{blue}{-13}&\color{blue}{53}&\color{orangered}{-211} \end{array} $$The solution is:
$$ \frac{ 2x^{3}-5x^{2}+x+1 }{ x+4 } = \color{blue}{2x^{2}-13x+53} \color{red}{~-~} \frac{ \color{red}{ 211 } }{ x+4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&2&-5&1&1\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-4&\color{orangered}{ 2 }&-5&1&1\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 2 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&2&-5&1&1\\& & \color{blue}{-8} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -13 } $
$$ \begin{array}{c|rrrr}-4&2&\color{orangered}{ -5 }&1&1\\& & \color{orangered}{-8} & & \\ \hline &2&\color{orangered}{-13}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -13 \right) } = \color{blue}{ 52 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&2&-5&1&1\\& & -8& \color{blue}{52} & \\ \hline &2&\color{blue}{-13}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 52 } = \color{orangered}{ 53 } $
$$ \begin{array}{c|rrrr}-4&2&-5&\color{orangered}{ 1 }&1\\& & -8& \color{orangered}{52} & \\ \hline &2&-13&\color{orangered}{53}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 53 } = \color{blue}{ -212 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&2&-5&1&1\\& & -8& 52& \color{blue}{-212} \\ \hline &2&-13&\color{blue}{53}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -212 \right) } = \color{orangered}{ -211 } $
$$ \begin{array}{c|rrrr}-4&2&-5&1&\color{orangered}{ 1 }\\& & -8& 52& \color{orangered}{-212} \\ \hline &\color{blue}{2}&\color{blue}{-13}&\color{blue}{53}&\color{orangered}{-211} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}-13x+53 } $ with a remainder of $ \color{red}{ -211 } $.