Divide using synthetic division $$ ( 2x^{4}+13x^{3}-2x^{2}-73x+60 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-5&2&13&-2&-73&60\\& & -10& -15& 85& \color{black}{-60} \\ \hline &\color{blue}{2}&\color{blue}{3}&\color{blue}{-17}&\color{blue}{12}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 2x^{4}+13x^{3}-2x^{2}-73x+60 }{ x+5 } = \color{blue}{2x^{3}+3x^{2}-17x+12} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&2&13&-2&-73&60\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-5&\color{orangered}{ 2 }&13&-2&-73&60\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 2 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&2&13&-2&-73&60\\& & \color{blue}{-10} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 13 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrr}-5&2&\color{orangered}{ 13 }&-2&-73&60\\& & \color{orangered}{-10} & & & \\ \hline &2&\color{orangered}{3}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 3 } = \color{blue}{ -15 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&2&13&-2&-73&60\\& & -10& \color{blue}{-15} & & \\ \hline &2&\color{blue}{3}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrrrr}-5&2&13&\color{orangered}{ -2 }&-73&60\\& & -10& \color{orangered}{-15} & & \\ \hline &2&3&\color{orangered}{-17}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ 85 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&2&13&-2&-73&60\\& & -10& -15& \color{blue}{85} & \\ \hline &2&3&\color{blue}{-17}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -73 } + \color{orangered}{ 85 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrrrr}-5&2&13&-2&\color{orangered}{ -73 }&60\\& & -10& -15& \color{orangered}{85} & \\ \hline &2&3&-17&\color{orangered}{12}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 12 } = \color{blue}{ -60 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&2&13&-2&-73&60\\& & -10& -15& 85& \color{blue}{-60} \\ \hline &2&3&-17&\color{blue}{12}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 60 } + \color{orangered}{ \left( -60 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-5&2&13&-2&-73&\color{orangered}{ 60 }\\& & -10& -15& 85& \color{orangered}{-60} \\ \hline &\color{blue}{2}&\color{blue}{3}&\color{blue}{-17}&\color{blue}{12}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}+3x^{2}-17x+12 } $ with a remainder of $ \color{red}{ 0 } $.