Divide using synthetic division $$ ( 3x^{2}+26x+40 ) \div ( x+7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-7&3&26&40\\& & -21& \color{black}{-35} \\ \hline &\color{blue}{3}&\color{blue}{5}&\color{orangered}{5} \end{array} $$The solution is:
$$ \frac{ 3x^{2}+26x+40 }{ x+7 } = \color{blue}{3x+5} ~+~ \frac{ \color{red}{ 5 } }{ x+7 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 7 = 0 $ ( $ x = \color{blue}{ -7 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-7}&3&26&40\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-7&\color{orangered}{ 3 }&26&40\\& & & \\ \hline &\color{orangered}{3}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 3 } = \color{blue}{ -21 } $.
$$ \begin{array}{c|rrr}\color{blue}{-7}&3&26&40\\& & \color{blue}{-21} & \\ \hline &\color{blue}{3}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 26 } + \color{orangered}{ \left( -21 \right) } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrr}-7&3&\color{orangered}{ 26 }&40\\& & \color{orangered}{-21} & \\ \hline &3&\color{orangered}{5}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 5 } = \color{blue}{ -35 } $.
$$ \begin{array}{c|rrr}\color{blue}{-7}&3&26&40\\& & -21& \color{blue}{-35} \\ \hline &3&\color{blue}{5}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 40 } + \color{orangered}{ \left( -35 \right) } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrr}-7&3&26&\color{orangered}{ 40 }\\& & -21& \color{orangered}{-35} \\ \hline &\color{blue}{3}&\color{blue}{5}&\color{orangered}{5} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x+5 } $ with a remainder of $ \color{red}{ 5 } $.