Divide using synthetic division $$ ( 5x^{2}+2x+3 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-2&5&2&3\\& & -10& \color{black}{16} \\ \hline &\color{blue}{5}&\color{blue}{-8}&\color{orangered}{19} \end{array} $$The solution is:
$$ \frac{ 5x^{2}+2x+3 }{ x+2 } = \color{blue}{5x-8} ~+~ \frac{ \color{red}{ 19 } }{ x+2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-2}&5&2&3\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-2&\color{orangered}{ 5 }&2&3\\& & & \\ \hline &\color{orangered}{5}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 5 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&5&2&3\\& & \color{blue}{-10} & \\ \hline &\color{blue}{5}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrr}-2&5&\color{orangered}{ 2 }&3\\& & \color{orangered}{-10} & \\ \hline &5&\color{orangered}{-8}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&5&2&3\\& & -10& \color{blue}{16} \\ \hline &5&\color{blue}{-8}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 16 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrr}-2&5&2&\color{orangered}{ 3 }\\& & -10& \color{orangered}{16} \\ \hline &\color{blue}{5}&\color{blue}{-8}&\color{orangered}{19} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x-8 } $ with a remainder of $ \color{red}{ 19 } $.