Divide using synthetic division $$ ( 4x^{4}-28x+9 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-2&4&0&0&-28&9\\& & -8& 16& -32& \color{black}{120} \\ \hline &\color{blue}{4}&\color{blue}{-8}&\color{blue}{16}&\color{blue}{-60}&\color{orangered}{129} \end{array} $$The solution is:
$$ \frac{ 4x^{4}-28x+9 }{ x+2 } = \color{blue}{4x^{3}-8x^{2}+16x-60} ~+~ \frac{ \color{red}{ 129 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&4&0&0&-28&9\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-2&\color{orangered}{ 4 }&0&0&-28&9\\& & & & & \\ \hline &\color{orangered}{4}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&4&0&0&-28&9\\& & \color{blue}{-8} & & & \\ \hline &\color{blue}{4}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrr}-2&4&\color{orangered}{ 0 }&0&-28&9\\& & \color{orangered}{-8} & & & \\ \hline &4&\color{orangered}{-8}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&4&0&0&-28&9\\& & -8& \color{blue}{16} & & \\ \hline &4&\color{blue}{-8}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 16 } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrrrr}-2&4&0&\color{orangered}{ 0 }&-28&9\\& & -8& \color{orangered}{16} & & \\ \hline &4&-8&\color{orangered}{16}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 16 } = \color{blue}{ -32 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&4&0&0&-28&9\\& & -8& 16& \color{blue}{-32} & \\ \hline &4&-8&\color{blue}{16}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -28 } + \color{orangered}{ \left( -32 \right) } = \color{orangered}{ -60 } $
$$ \begin{array}{c|rrrrr}-2&4&0&0&\color{orangered}{ -28 }&9\\& & -8& 16& \color{orangered}{-32} & \\ \hline &4&-8&16&\color{orangered}{-60}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -60 \right) } = \color{blue}{ 120 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&4&0&0&-28&9\\& & -8& 16& -32& \color{blue}{120} \\ \hline &4&-8&16&\color{blue}{-60}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 120 } = \color{orangered}{ 129 } $
$$ \begin{array}{c|rrrrr}-2&4&0&0&-28&\color{orangered}{ 9 }\\& & -8& 16& -32& \color{orangered}{120} \\ \hline &\color{blue}{4}&\color{blue}{-8}&\color{blue}{16}&\color{blue}{-60}&\color{orangered}{129} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{3}-8x^{2}+16x-60 } $ with a remainder of $ \color{red}{ 129 } $.