Divide using synthetic division $$ ( 5x^{2}+3x-22 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-3&5&3&-22\\& & -15& \color{black}{36} \\ \hline &\color{blue}{5}&\color{blue}{-12}&\color{orangered}{14} \end{array} $$The solution is:
$$ \frac{ 5x^{2}+3x-22 }{ x+3 } = \color{blue}{5x-12} ~+~ \frac{ \color{red}{ 14 } }{ x+3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-3}&5&3&-22\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-3&\color{orangered}{ 5 }&3&-22\\& & & \\ \hline &\color{orangered}{5}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 5 } = \color{blue}{ -15 } $.
$$ \begin{array}{c|rrr}\color{blue}{-3}&5&3&-22\\& & \color{blue}{-15} & \\ \hline &\color{blue}{5}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrr}-3&5&\color{orangered}{ 3 }&-22\\& & \color{orangered}{-15} & \\ \hline &5&\color{orangered}{-12}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrr}\color{blue}{-3}&5&3&-22\\& & -15& \color{blue}{36} \\ \hline &5&\color{blue}{-12}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -22 } + \color{orangered}{ 36 } = \color{orangered}{ 14 } $
$$ \begin{array}{c|rrr}-3&5&3&\color{orangered}{ -22 }\\& & -15& \color{orangered}{36} \\ \hline &\color{blue}{5}&\color{blue}{-12}&\color{orangered}{14} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x-12 } $ with a remainder of $ \color{red}{ 14 } $.