Divide using synthetic division $$ ( 4x^{2}+5x-3 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-4&4&5&-3\\& & -16& \color{black}{44} \\ \hline &\color{blue}{4}&\color{blue}{-11}&\color{orangered}{41} \end{array} $$The solution is:
$$ \frac{ 4x^{2}+5x-3 }{ x+4 } = \color{blue}{4x-11} ~+~ \frac{ \color{red}{ 41 } }{ x+4 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-4}&4&5&-3\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-4&\color{orangered}{ 4 }&5&-3\\& & & \\ \hline &\color{orangered}{4}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 4 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&4&5&-3\\& & \color{blue}{-16} & \\ \hline &\color{blue}{4}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -11 } $
$$ \begin{array}{c|rrr}-4&4&\color{orangered}{ 5 }&-3\\& & \color{orangered}{-16} & \\ \hline &4&\color{orangered}{-11}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -11 \right) } = \color{blue}{ 44 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&4&5&-3\\& & -16& \color{blue}{44} \\ \hline &4&\color{blue}{-11}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 44 } = \color{orangered}{ 41 } $
$$ \begin{array}{c|rrr}-4&4&5&\color{orangered}{ -3 }\\& & -16& \color{orangered}{44} \\ \hline &\color{blue}{4}&\color{blue}{-11}&\color{orangered}{41} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x-11 } $ with a remainder of $ \color{red}{ 41 } $.