Divide using synthetic division $$ ( 6x^{3}-5x^{2}+1 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-4&6&-5&0&1\\& & -24& 116& \color{black}{-464} \\ \hline &\color{blue}{6}&\color{blue}{-29}&\color{blue}{116}&\color{orangered}{-463} \end{array} $$The solution is:
$$ \frac{ 6x^{3}-5x^{2}+1 }{ x+4 } = \color{blue}{6x^{2}-29x+116} \color{red}{~-~} \frac{ \color{red}{ 463 } }{ x+4 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&6&-5&0&1\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-4&\color{orangered}{ 6 }&-5&0&1\\& & & & \\ \hline &\color{orangered}{6}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 6 } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&6&-5&0&1\\& & \color{blue}{-24} & & \\ \hline &\color{blue}{6}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -29 } $
$$ \begin{array}{c|rrrr}-4&6&\color{orangered}{ -5 }&0&1\\& & \color{orangered}{-24} & & \\ \hline &6&\color{orangered}{-29}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -29 \right) } = \color{blue}{ 116 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&6&-5&0&1\\& & -24& \color{blue}{116} & \\ \hline &6&\color{blue}{-29}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 116 } = \color{orangered}{ 116 } $
$$ \begin{array}{c|rrrr}-4&6&-5&\color{orangered}{ 0 }&1\\& & -24& \color{orangered}{116} & \\ \hline &6&-29&\color{orangered}{116}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 116 } = \color{blue}{ -464 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&6&-5&0&1\\& & -24& 116& \color{blue}{-464} \\ \hline &6&-29&\color{blue}{116}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -464 \right) } = \color{orangered}{ -463 } $
$$ \begin{array}{c|rrrr}-4&6&-5&0&\color{orangered}{ 1 }\\& & -24& 116& \color{orangered}{-464} \\ \hline &\color{blue}{6}&\color{blue}{-29}&\color{blue}{116}&\color{orangered}{-463} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{2}-29x+116 } $ with a remainder of $ \color{red}{ -463 } $.