Divide using synthetic division $$ ( 3x^{6}-75x^{4}+3x^{2}-90 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-5&3&0&-75&0&3&0&-90\\& & -15& 75& 0& 0& -15& \color{black}{75} \\ \hline &\color{blue}{3}&\color{blue}{-15}&\color{blue}{0}&\color{blue}{0}&\color{blue}{3}&\color{blue}{-15}&\color{orangered}{-15} \end{array} $$The solution is:
$$ \frac{ 3x^{6}-75x^{4}+3x^{2}-90 }{ x+5 } = \color{blue}{3x^{5}-15x^{4}+3x-15} \color{red}{~-~} \frac{ \color{red}{ 15 } }{ x+5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&3&0&-75&0&3&0&-90\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-5&\color{orangered}{ 3 }&0&-75&0&3&0&-90\\& & & & & & & \\ \hline &\color{orangered}{3}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 3 } = \color{blue}{ -15 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&3&0&-75&0&3&0&-90\\& & \color{blue}{-15} & & & & & \\ \hline &\color{blue}{3}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ -15 } $
$$ \begin{array}{c|rrrrrrr}-5&3&\color{orangered}{ 0 }&-75&0&3&0&-90\\& & \color{orangered}{-15} & & & & & \\ \hline &3&\color{orangered}{-15}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -15 \right) } = \color{blue}{ 75 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&3&0&-75&0&3&0&-90\\& & -15& \color{blue}{75} & & & & \\ \hline &3&\color{blue}{-15}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -75 } + \color{orangered}{ 75 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrr}-5&3&0&\color{orangered}{ -75 }&0&3&0&-90\\& & -15& \color{orangered}{75} & & & & \\ \hline &3&-15&\color{orangered}{0}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&3&0&-75&0&3&0&-90\\& & -15& 75& \color{blue}{0} & & & \\ \hline &3&-15&\color{blue}{0}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrr}-5&3&0&-75&\color{orangered}{ 0 }&3&0&-90\\& & -15& 75& \color{orangered}{0} & & & \\ \hline &3&-15&0&\color{orangered}{0}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&3&0&-75&0&3&0&-90\\& & -15& 75& 0& \color{blue}{0} & & \\ \hline &3&-15&0&\color{blue}{0}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 0 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrrrr}-5&3&0&-75&0&\color{orangered}{ 3 }&0&-90\\& & -15& 75& 0& \color{orangered}{0} & & \\ \hline &3&-15&0&0&\color{orangered}{3}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 3 } = \color{blue}{ -15 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&3&0&-75&0&3&0&-90\\& & -15& 75& 0& 0& \color{blue}{-15} & \\ \hline &3&-15&0&0&\color{blue}{3}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ -15 } $
$$ \begin{array}{c|rrrrrrr}-5&3&0&-75&0&3&\color{orangered}{ 0 }&-90\\& & -15& 75& 0& 0& \color{orangered}{-15} & \\ \hline &3&-15&0&0&3&\color{orangered}{-15}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -15 \right) } = \color{blue}{ 75 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&3&0&-75&0&3&0&-90\\& & -15& 75& 0& 0& -15& \color{blue}{75} \\ \hline &3&-15&0&0&3&\color{blue}{-15}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ -90 } + \color{orangered}{ 75 } = \color{orangered}{ -15 } $
$$ \begin{array}{c|rrrrrrr}-5&3&0&-75&0&3&0&\color{orangered}{ -90 }\\& & -15& 75& 0& 0& -15& \color{orangered}{75} \\ \hline &\color{blue}{3}&\color{blue}{-15}&\color{blue}{0}&\color{blue}{0}&\color{blue}{3}&\color{blue}{-15}&\color{orangered}{-15} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{5}-15x^{4}+3x-15 } $ with a remainder of $ \color{red}{ -15 } $.