Divide using synthetic division $$ ( 4x^{2}-3 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-5&4&0&-3\\& & -20& \color{black}{100} \\ \hline &\color{blue}{4}&\color{blue}{-20}&\color{orangered}{97} \end{array} $$The solution is:
$$ \frac{ 4x^{2}-3 }{ x+5 } = \color{blue}{4x-20} ~+~ \frac{ \color{red}{ 97 } }{ x+5 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-5}&4&0&-3\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-5&\color{orangered}{ 4 }&0&-3\\& & & \\ \hline &\color{orangered}{4}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&4&0&-3\\& & \color{blue}{-20} & \\ \hline &\color{blue}{4}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ -20 } $
$$ \begin{array}{c|rrr}-5&4&\color{orangered}{ 0 }&-3\\& & \color{orangered}{-20} & \\ \hline &4&\color{orangered}{-20}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -20 \right) } = \color{blue}{ 100 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&4&0&-3\\& & -20& \color{blue}{100} \\ \hline &4&\color{blue}{-20}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 100 } = \color{orangered}{ 97 } $
$$ \begin{array}{c|rrr}-5&4&0&\color{orangered}{ -3 }\\& & -20& \color{orangered}{100} \\ \hline &\color{blue}{4}&\color{blue}{-20}&\color{orangered}{97} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x-20 } $ with a remainder of $ \color{red}{ 97 } $.