Divide using synthetic division $$ ( 2x^{2}+3x-5 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}4&2&3&-5\\& & 8& \color{black}{44} \\ \hline &\color{blue}{2}&\color{blue}{11}&\color{orangered}{39} \end{array} $$The solution is:
$$ \frac{ 2x^{2}+3x-5 }{ x-4 } = \color{blue}{2x+11} ~+~ \frac{ \color{red}{ 39 } }{ x-4 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{4}&2&3&-5\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}4&\color{orangered}{ 2 }&3&-5\\& & & \\ \hline &\color{orangered}{2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 2 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&2&3&-5\\& & \color{blue}{8} & \\ \hline &\color{blue}{2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 8 } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrr}4&2&\color{orangered}{ 3 }&-5\\& & \color{orangered}{8} & \\ \hline &2&\color{orangered}{11}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 11 } = \color{blue}{ 44 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&2&3&-5\\& & 8& \color{blue}{44} \\ \hline &2&\color{blue}{11}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 44 } = \color{orangered}{ 39 } $
$$ \begin{array}{c|rrr}4&2&3&\color{orangered}{ -5 }\\& & 8& \color{orangered}{44} \\ \hline &\color{blue}{2}&\color{blue}{11}&\color{orangered}{39} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x+11 } $ with a remainder of $ \color{red}{ 39 } $.