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