Divide using synthetic division $$ ( 3x^{4}-x^{3}-88x^{2}+126x-180 ) \div ( x+6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-6&3&-1&-88&126&-180\\& & -18& 114& -156& \color{black}{180} \\ \hline &\color{blue}{3}&\color{blue}{-19}&\color{blue}{26}&\color{blue}{-30}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 3x^{4}-x^{3}-88x^{2}+126x-180 }{ x+6 } = \color{blue}{3x^{3}-19x^{2}+26x-30} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 6 = 0 $ ( $ x = \color{blue}{ -6 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&3&-1&-88&126&-180\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-6&\color{orangered}{ 3 }&-1&-88&126&-180\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 3 } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&3&-1&-88&126&-180\\& & \color{blue}{-18} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -19 } $
$$ \begin{array}{c|rrrrr}-6&3&\color{orangered}{ -1 }&-88&126&-180\\& & \color{orangered}{-18} & & & \\ \hline &3&\color{orangered}{-19}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -19 \right) } = \color{blue}{ 114 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&3&-1&-88&126&-180\\& & -18& \color{blue}{114} & & \\ \hline &3&\color{blue}{-19}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -88 } + \color{orangered}{ 114 } = \color{orangered}{ 26 } $
$$ \begin{array}{c|rrrrr}-6&3&-1&\color{orangered}{ -88 }&126&-180\\& & -18& \color{orangered}{114} & & \\ \hline &3&-19&\color{orangered}{26}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 26 } = \color{blue}{ -156 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&3&-1&-88&126&-180\\& & -18& 114& \color{blue}{-156} & \\ \hline &3&-19&\color{blue}{26}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 126 } + \color{orangered}{ \left( -156 \right) } = \color{orangered}{ -30 } $
$$ \begin{array}{c|rrrrr}-6&3&-1&-88&\color{orangered}{ 126 }&-180\\& & -18& 114& \color{orangered}{-156} & \\ \hline &3&-19&26&\color{orangered}{-30}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -30 \right) } = \color{blue}{ 180 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&3&-1&-88&126&-180\\& & -18& 114& -156& \color{blue}{180} \\ \hline &3&-19&26&\color{blue}{-30}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -180 } + \color{orangered}{ 180 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-6&3&-1&-88&126&\color{orangered}{ -180 }\\& & -18& 114& -156& \color{orangered}{180} \\ \hline &\color{blue}{3}&\color{blue}{-19}&\color{blue}{26}&\color{blue}{-30}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}-19x^{2}+26x-30 } $ with a remainder of $ \color{red}{ 0 } $.