Divide using synthetic division $$ ( x^{2}+8x+1 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}4&1&8&1\\& & 4& \color{black}{48} \\ \hline &\color{blue}{1}&\color{blue}{12}&\color{orangered}{49} \end{array} $$The solution is:
$$ \frac{ x^{2}+8x+1 }{ x-4 } = \color{blue}{x+12} ~+~ \frac{ \color{red}{ 49 } }{ x-4 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{4}&1&8&1\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}4&\color{orangered}{ 1 }&8&1\\& & & \\ \hline &\color{orangered}{1}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 1 } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&1&8&1\\& & \color{blue}{4} & \\ \hline &\color{blue}{1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 4 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrr}4&1&\color{orangered}{ 8 }&1\\& & \color{orangered}{4} & \\ \hline &1&\color{orangered}{12}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 12 } = \color{blue}{ 48 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&1&8&1\\& & 4& \color{blue}{48} \\ \hline &1&\color{blue}{12}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 48 } = \color{orangered}{ 49 } $
$$ \begin{array}{c|rrr}4&1&8&\color{orangered}{ 1 }\\& & 4& \color{orangered}{48} \\ \hline &\color{blue}{1}&\color{blue}{12}&\color{orangered}{49} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x+12 } $ with a remainder of $ \color{red}{ 49 } $.