Divide using synthetic division $$ ( 4x^{2}+3x-32 ) \div ( 0 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}0&4&3&-32\\& & 0& \color{black}{0} \\ \hline &\color{blue}{4}&\color{blue}{3}&\color{orangered}{-32} \end{array} $$The solution is:
$$ \frac{ 4x^{2}+3x-32 }{ x } = \color{blue}{4x+3} \color{red}{~-~} \frac{ \color{red}{ 32 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrr}\color{blue}{0}&4&3&-32\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}0&\color{orangered}{ 4 }&3&-32\\& & & \\ \hline &\color{orangered}{4}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 4 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrr}\color{blue}{0}&4&3&-32\\& & \color{blue}{0} & \\ \hline &\color{blue}{4}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 0 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrr}0&4&\color{orangered}{ 3 }&-32\\& & \color{orangered}{0} & \\ \hline &4&\color{orangered}{3}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 3 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrr}\color{blue}{0}&4&3&-32\\& & 0& \color{blue}{0} \\ \hline &4&\color{blue}{3}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -32 } + \color{orangered}{ 0 } = \color{orangered}{ -32 } $
$$ \begin{array}{c|rrr}0&4&3&\color{orangered}{ -32 }\\& & 0& \color{orangered}{0} \\ \hline &\color{blue}{4}&\color{blue}{3}&\color{orangered}{-32} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x+3 } $ with a remainder of $ \color{red}{ -32 } $.