Divide using synthetic division $$ ( 15x^{2}+14x-8 ) \div ( 3x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}4&15&14&-8\\& & 60& \color{black}{296} \\ \hline &\color{blue}{15}&\color{blue}{74}&\color{orangered}{288} \end{array} $$The solution is:
$$ \frac{ 15x^{2}+14x-8 }{ x-4 } = \color{blue}{15x+74} ~+~ \frac{ \color{red}{ 288 } }{ x-4 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{4}&15&14&-8\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}4&\color{orangered}{ 15 }&14&-8\\& & & \\ \hline &\color{orangered}{15}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 15 } = \color{blue}{ 60 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&15&14&-8\\& & \color{blue}{60} & \\ \hline &\color{blue}{15}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ 60 } = \color{orangered}{ 74 } $
$$ \begin{array}{c|rrr}4&15&\color{orangered}{ 14 }&-8\\& & \color{orangered}{60} & \\ \hline &15&\color{orangered}{74}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 74 } = \color{blue}{ 296 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&15&14&-8\\& & 60& \color{blue}{296} \\ \hline &15&\color{blue}{74}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 296 } = \color{orangered}{ 288 } $
$$ \begin{array}{c|rrr}4&15&14&\color{orangered}{ -8 }\\& & 60& \color{orangered}{296} \\ \hline &\color{blue}{15}&\color{blue}{74}&\color{orangered}{288} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 15x+74 } $ with a remainder of $ \color{red}{ 288 } $.