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