Divide using synthetic division $$ ( 4x^{2}+5x-4 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}4&4&5&-4\\& & 16& \color{black}{84} \\ \hline &\color{blue}{4}&\color{blue}{21}&\color{orangered}{80} \end{array} $$The solution is:
$$ \frac{ 4x^{2}+5x-4 }{ x-4 } = \color{blue}{4x+21} ~+~ \frac{ \color{red}{ 80 } }{ x-4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -4 = 0 $ ( $ x = \color{blue}{ 4 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{4}&4&5&-4\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}4&\color{orangered}{ 4 }&5&-4\\& & & \\ \hline &\color{orangered}{4}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 4 } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&4&5&-4\\& & \color{blue}{16} & \\ \hline &\color{blue}{4}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 16 } = \color{orangered}{ 21 } $
$$ \begin{array}{c|rrr}4&4&\color{orangered}{ 5 }&-4\\& & \color{orangered}{16} & \\ \hline &4&\color{orangered}{21}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 21 } = \color{blue}{ 84 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&4&5&-4\\& & 16& \color{blue}{84} \\ \hline &4&\color{blue}{21}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 84 } = \color{orangered}{ 80 } $
$$ \begin{array}{c|rrr}4&4&5&\color{orangered}{ -4 }\\& & 16& \color{orangered}{84} \\ \hline &\color{blue}{4}&\color{blue}{21}&\color{orangered}{80} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x+21 } $ with a remainder of $ \color{red}{ 80 } $.