Divide using synthetic division $$ ( 6x^{4}+45x-6 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-2&6&0&0&45&-6\\& & -12& 24& -48& \color{black}{6} \\ \hline &\color{blue}{6}&\color{blue}{-12}&\color{blue}{24}&\color{blue}{-3}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 6x^{4}+45x-6 }{ x+2 } = \color{blue}{6x^{3}-12x^{2}+24x-3} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&6&0&0&45&-6\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-2&\color{orangered}{ 6 }&0&0&45&-6\\& & & & & \\ \hline &\color{orangered}{6}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 6 } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&6&0&0&45&-6\\& & \color{blue}{-12} & & & \\ \hline &\color{blue}{6}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrrr}-2&6&\color{orangered}{ 0 }&0&45&-6\\& & \color{orangered}{-12} & & & \\ \hline &6&\color{orangered}{-12}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&6&0&0&45&-6\\& & -12& \color{blue}{24} & & \\ \hline &6&\color{blue}{-12}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 24 } = \color{orangered}{ 24 } $
$$ \begin{array}{c|rrrrr}-2&6&0&\color{orangered}{ 0 }&45&-6\\& & -12& \color{orangered}{24} & & \\ \hline &6&-12&\color{orangered}{24}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 24 } = \color{blue}{ -48 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&6&0&0&45&-6\\& & -12& 24& \color{blue}{-48} & \\ \hline &6&-12&\color{blue}{24}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 45 } + \color{orangered}{ \left( -48 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrr}-2&6&0&0&\color{orangered}{ 45 }&-6\\& & -12& 24& \color{orangered}{-48} & \\ \hline &6&-12&24&\color{orangered}{-3}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&6&0&0&45&-6\\& & -12& 24& -48& \color{blue}{6} \\ \hline &6&-12&24&\color{blue}{-3}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 6 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-2&6&0&0&45&\color{orangered}{ -6 }\\& & -12& 24& -48& \color{orangered}{6} \\ \hline &\color{blue}{6}&\color{blue}{-12}&\color{blue}{24}&\color{blue}{-3}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{3}-12x^{2}+24x-3 } $ with a remainder of $ \color{red}{ 0 } $.