Divide using synthetic division $$ ( 4x^{3}+21x^{2}+26x+320 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-5&4&21&26&320\\& & -20& -5& \color{black}{-105} \\ \hline &\color{blue}{4}&\color{blue}{1}&\color{blue}{21}&\color{orangered}{215} \end{array} $$The solution is:
$$ \frac{ 4x^{3}+21x^{2}+26x+320 }{ x+5 } = \color{blue}{4x^{2}+x+21} ~+~ \frac{ \color{red}{ 215 } }{ x+5 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&4&21&26&320\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-5&\color{orangered}{ 4 }&21&26&320\\& & & & \\ \hline &\color{orangered}{4}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 4 } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&4&21&26&320\\& & \color{blue}{-20} & & \\ \hline &\color{blue}{4}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrr}-5&4&\color{orangered}{ 21 }&26&320\\& & \color{orangered}{-20} & & \\ \hline &4&\color{orangered}{1}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 1 } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&4&21&26&320\\& & -20& \color{blue}{-5} & \\ \hline &4&\color{blue}{1}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 26 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ 21 } $
$$ \begin{array}{c|rrrr}-5&4&21&\color{orangered}{ 26 }&320\\& & -20& \color{orangered}{-5} & \\ \hline &4&1&\color{orangered}{21}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 21 } = \color{blue}{ -105 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&4&21&26&320\\& & -20& -5& \color{blue}{-105} \\ \hline &4&1&\color{blue}{21}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 320 } + \color{orangered}{ \left( -105 \right) } = \color{orangered}{ 215 } $
$$ \begin{array}{c|rrrr}-5&4&21&26&\color{orangered}{ 320 }\\& & -20& -5& \color{orangered}{-105} \\ \hline &\color{blue}{4}&\color{blue}{1}&\color{blue}{21}&\color{orangered}{215} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{2}+x+21 } $ with a remainder of $ \color{red}{ 215 } $.