Divide using synthetic division $$ ( 3x^{4}-16x^{3}-7x^{2}+64x-20 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-5&3&-16&-7&64&-20\\& & -15& 155& -740& \color{black}{3380} \\ \hline &\color{blue}{3}&\color{blue}{-31}&\color{blue}{148}&\color{blue}{-676}&\color{orangered}{3360} \end{array} $$The solution is:
$$ \frac{ 3x^{4}-16x^{3}-7x^{2}+64x-20 }{ x+5 } = \color{blue}{3x^{3}-31x^{2}+148x-676} ~+~ \frac{ \color{red}{ 3360 } }{ x+5 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&3&-16&-7&64&-20\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-5&\color{orangered}{ 3 }&-16&-7&64&-20\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&3&-16&-7&64&-20\\& & \color{blue}{-15} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ -31 } $
$$ \begin{array}{c|rrrrr}-5&3&\color{orangered}{ -16 }&-7&64&-20\\& & \color{orangered}{-15} & & & \\ \hline &3&\color{orangered}{-31}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -31 \right) } = \color{blue}{ 155 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&3&-16&-7&64&-20\\& & -15& \color{blue}{155} & & \\ \hline &3&\color{blue}{-31}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 155 } = \color{orangered}{ 148 } $
$$ \begin{array}{c|rrrrr}-5&3&-16&\color{orangered}{ -7 }&64&-20\\& & -15& \color{orangered}{155} & & \\ \hline &3&-31&\color{orangered}{148}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 148 } = \color{blue}{ -740 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&3&-16&-7&64&-20\\& & -15& 155& \color{blue}{-740} & \\ \hline &3&-31&\color{blue}{148}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 64 } + \color{orangered}{ \left( -740 \right) } = \color{orangered}{ -676 } $
$$ \begin{array}{c|rrrrr}-5&3&-16&-7&\color{orangered}{ 64 }&-20\\& & -15& 155& \color{orangered}{-740} & \\ \hline &3&-31&148&\color{orangered}{-676}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -676 \right) } = \color{blue}{ 3380 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&3&-16&-7&64&-20\\& & -15& 155& -740& \color{blue}{3380} \\ \hline &3&-31&148&\color{blue}{-676}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 3380 } = \color{orangered}{ 3360 } $
$$ \begin{array}{c|rrrrr}-5&3&-16&-7&64&\color{orangered}{ -20 }\\& & -15& 155& -740& \color{orangered}{3380} \\ \hline &\color{blue}{3}&\color{blue}{-31}&\color{blue}{148}&\color{blue}{-676}&\color{orangered}{3360} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}-31x^{2}+148x-676 } $ with a remainder of $ \color{red}{ 3360 } $.