Divide using synthetic division $$ ( 8x^{3}-x^{2}-12x+1 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-4&8&-1&-12&1\\& & -32& 132& \color{black}{-480} \\ \hline &\color{blue}{8}&\color{blue}{-33}&\color{blue}{120}&\color{orangered}{-479} \end{array} $$The solution is:
$$ \frac{ 8x^{3}-x^{2}-12x+1 }{ x+4 } = \color{blue}{8x^{2}-33x+120} \color{red}{~-~} \frac{ \color{red}{ 479 } }{ x+4 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&8&-1&-12&1\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-4&\color{orangered}{ 8 }&-1&-12&1\\& & & & \\ \hline &\color{orangered}{8}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 8 } = \color{blue}{ -32 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&8&-1&-12&1\\& & \color{blue}{-32} & & \\ \hline &\color{blue}{8}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -32 \right) } = \color{orangered}{ -33 } $
$$ \begin{array}{c|rrrr}-4&8&\color{orangered}{ -1 }&-12&1\\& & \color{orangered}{-32} & & \\ \hline &8&\color{orangered}{-33}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -33 \right) } = \color{blue}{ 132 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&8&-1&-12&1\\& & -32& \color{blue}{132} & \\ \hline &8&\color{blue}{-33}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 132 } = \color{orangered}{ 120 } $
$$ \begin{array}{c|rrrr}-4&8&-1&\color{orangered}{ -12 }&1\\& & -32& \color{orangered}{132} & \\ \hline &8&-33&\color{orangered}{120}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 120 } = \color{blue}{ -480 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&8&-1&-12&1\\& & -32& 132& \color{blue}{-480} \\ \hline &8&-33&\color{blue}{120}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -480 \right) } = \color{orangered}{ -479 } $
$$ \begin{array}{c|rrrr}-4&8&-1&-12&\color{orangered}{ 1 }\\& & -32& 132& \color{orangered}{-480} \\ \hline &\color{blue}{8}&\color{blue}{-33}&\color{blue}{120}&\color{orangered}{-479} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{2}-33x+120 } $ with a remainder of $ \color{red}{ -479 } $.