Divide using synthetic division $$ ( 2x^{2}-3x+4 ) \div ( x+5 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrr}-5&2&-3&4\\& & -10& \color{black}{65} \\ \hline &\color{blue}{2}&\color{blue}{-13}&\color{orangered}{69} \end{array} $$The solution is:
$$ \dfrac{ 2x^{2}-3x+4 }{ x+5 } = \color{blue}{2x-13} ~+~ \dfrac{ \color{red}{ 69 } }{ x+5 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-5}&2&-3&4\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-5&\color{orangered}{ 2 }&-3&4\\& & & \\ \hline &\color{orangered}{2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 2 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&2&-3&4\\& & \color{blue}{-10} & \\ \hline &\color{blue}{2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -13 } $
$$ \begin{array}{c|rrr}-5&2&\color{orangered}{ -3 }&4\\& & \color{orangered}{-10} & \\ \hline &2&\color{orangered}{-13}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -13 \right) } = \color{blue}{ 65 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&2&-3&4\\& & -10& \color{blue}{65} \\ \hline &2&\color{blue}{-13}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 65 } = \color{orangered}{ 69 } $
$$ \begin{array}{c|rrr}-5&2&-3&\color{orangered}{ 4 }\\& & -10& \color{orangered}{65} \\ \hline &\color{blue}{2}&\color{blue}{-13}&\color{orangered}{69} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x-13 } $ with a remainder of $ \color{red}{ 69 } $.