Divide using synthetic division $$ ( 2x^{2}-7x+3 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-2&2&-7&3\\& & -4& \color{black}{22} \\ \hline &\color{blue}{2}&\color{blue}{-11}&\color{orangered}{25} \end{array} $$The solution is:
$$ \frac{ 2x^{2}-7x+3 }{ x+2 } = \color{blue}{2x-11} ~+~ \frac{ \color{red}{ 25 } }{ x+2 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-2}&2&-7&3\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-2&\color{orangered}{ 2 }&-7&3\\& & & \\ \hline &\color{orangered}{2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 2 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&2&-7&3\\& & \color{blue}{-4} & \\ \hline &\color{blue}{2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -11 } $
$$ \begin{array}{c|rrr}-2&2&\color{orangered}{ -7 }&3\\& & \color{orangered}{-4} & \\ \hline &2&\color{orangered}{-11}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -11 \right) } = \color{blue}{ 22 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&2&-7&3\\& & -4& \color{blue}{22} \\ \hline &2&\color{blue}{-11}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 22 } = \color{orangered}{ 25 } $
$$ \begin{array}{c|rrr}-2&2&-7&\color{orangered}{ 3 }\\& & -4& \color{orangered}{22} \\ \hline &\color{blue}{2}&\color{blue}{-11}&\color{orangered}{25} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x-11 } $ with a remainder of $ \color{red}{ 25 } $.