Divide using synthetic division $$ ( 7x^{2}-12x+5 ) \div ( 3x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}4&7&-12&5\\& & 28& \color{black}{64} \\ \hline &\color{blue}{7}&\color{blue}{16}&\color{orangered}{69} \end{array} $$The solution is:
$$ \frac{ 7x^{2}-12x+5 }{ x-4 } = \color{blue}{7x+16} ~+~ \frac{ \color{red}{ 69 } }{ x-4 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{4}&7&-12&5\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}4&\color{orangered}{ 7 }&-12&5\\& & & \\ \hline &\color{orangered}{7}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 7 } = \color{blue}{ 28 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&7&-12&5\\& & \color{blue}{28} & \\ \hline &\color{blue}{7}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 28 } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrr}4&7&\color{orangered}{ -12 }&5\\& & \color{orangered}{28} & \\ \hline &7&\color{orangered}{16}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 16 } = \color{blue}{ 64 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&7&-12&5\\& & 28& \color{blue}{64} \\ \hline &7&\color{blue}{16}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 64 } = \color{orangered}{ 69 } $
$$ \begin{array}{c|rrr}4&7&-12&\color{orangered}{ 5 }\\& & 28& \color{orangered}{64} \\ \hline &\color{blue}{7}&\color{blue}{16}&\color{orangered}{69} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 7x+16 } $ with a remainder of $ \color{red}{ 69 } $.