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