Divide using synthetic division $$ ( 4x^{4}-27x^{3}+19x^{2}-4x+48 ) \div ( 4x-7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}7&4&-27&19&-4&48\\& & 28& 7& 182& \color{black}{1246} \\ \hline &\color{blue}{4}&\color{blue}{1}&\color{blue}{26}&\color{blue}{178}&\color{orangered}{1294} \end{array} $$The solution is:
$$ \frac{ 4x^{4}-27x^{3}+19x^{2}-4x+48 }{ x-7 } = \color{blue}{4x^{3}+x^{2}+26x+178} ~+~ \frac{ \color{red}{ 1294 } }{ x-7 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -7 = 0 $ ( $ x = \color{blue}{ 7 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{7}&4&-27&19&-4&48\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}7&\color{orangered}{ 4 }&-27&19&-4&48\\& & & & & \\ \hline &\color{orangered}{4}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 4 } = \color{blue}{ 28 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{7}&4&-27&19&-4&48\\& & \color{blue}{28} & & & \\ \hline &\color{blue}{4}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -27 } + \color{orangered}{ 28 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrr}7&4&\color{orangered}{ -27 }&19&-4&48\\& & \color{orangered}{28} & & & \\ \hline &4&\color{orangered}{1}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 1 } = \color{blue}{ 7 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{7}&4&-27&19&-4&48\\& & 28& \color{blue}{7} & & \\ \hline &4&\color{blue}{1}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 19 } + \color{orangered}{ 7 } = \color{orangered}{ 26 } $
$$ \begin{array}{c|rrrrr}7&4&-27&\color{orangered}{ 19 }&-4&48\\& & 28& \color{orangered}{7} & & \\ \hline &4&1&\color{orangered}{26}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 26 } = \color{blue}{ 182 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{7}&4&-27&19&-4&48\\& & 28& 7& \color{blue}{182} & \\ \hline &4&1&\color{blue}{26}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 182 } = \color{orangered}{ 178 } $
$$ \begin{array}{c|rrrrr}7&4&-27&19&\color{orangered}{ -4 }&48\\& & 28& 7& \color{orangered}{182} & \\ \hline &4&1&26&\color{orangered}{178}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 178 } = \color{blue}{ 1246 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{7}&4&-27&19&-4&48\\& & 28& 7& 182& \color{blue}{1246} \\ \hline &4&1&26&\color{blue}{178}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 48 } + \color{orangered}{ 1246 } = \color{orangered}{ 1294 } $
$$ \begin{array}{c|rrrrr}7&4&-27&19&-4&\color{orangered}{ 48 }\\& & 28& 7& 182& \color{orangered}{1246} \\ \hline &\color{blue}{4}&\color{blue}{1}&\color{blue}{26}&\color{blue}{178}&\color{orangered}{1294} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{3}+x^{2}+26x+178 } $ with a remainder of $ \color{red}{ 1294 } $.