Divide using synthetic division $$ ( 4x^{4}+52x^{3}+87x^{2}-13x-22 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-2&4&52&87&-13&-22\\& & -8& -88& 2& \color{black}{22} \\ \hline &\color{blue}{4}&\color{blue}{44}&\color{blue}{-1}&\color{blue}{-11}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 4x^{4}+52x^{3}+87x^{2}-13x-22 }{ x+2 } = \color{blue}{4x^{3}+44x^{2}-x-11} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&4&52&87&-13&-22\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-2&\color{orangered}{ 4 }&52&87&-13&-22\\& & & & & \\ \hline &\color{orangered}{4}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 4 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&4&52&87&-13&-22\\& & \color{blue}{-8} & & & \\ \hline &\color{blue}{4}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 52 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ 44 } $
$$ \begin{array}{c|rrrrr}-2&4&\color{orangered}{ 52 }&87&-13&-22\\& & \color{orangered}{-8} & & & \\ \hline &4&\color{orangered}{44}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 44 } = \color{blue}{ -88 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&4&52&87&-13&-22\\& & -8& \color{blue}{-88} & & \\ \hline &4&\color{blue}{44}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 87 } + \color{orangered}{ \left( -88 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrr}-2&4&52&\color{orangered}{ 87 }&-13&-22\\& & -8& \color{orangered}{-88} & & \\ \hline &4&44&\color{orangered}{-1}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&4&52&87&-13&-22\\& & -8& -88& \color{blue}{2} & \\ \hline &4&44&\color{blue}{-1}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 2 } = \color{orangered}{ -11 } $
$$ \begin{array}{c|rrrrr}-2&4&52&87&\color{orangered}{ -13 }&-22\\& & -8& -88& \color{orangered}{2} & \\ \hline &4&44&-1&\color{orangered}{-11}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -11 \right) } = \color{blue}{ 22 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&4&52&87&-13&-22\\& & -8& -88& 2& \color{blue}{22} \\ \hline &4&44&-1&\color{blue}{-11}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -22 } + \color{orangered}{ 22 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-2&4&52&87&-13&\color{orangered}{ -22 }\\& & -8& -88& 2& \color{orangered}{22} \\ \hline &\color{blue}{4}&\color{blue}{44}&\color{blue}{-1}&\color{blue}{-11}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{3}+44x^{2}-x-11 } $ with a remainder of $ \color{red}{ 0 } $.