Divide using synthetic division $$ ( 10x^{7}+23x^{6}-13x^{5}+14x^{4}-20x^{3}+26x^{2}-14x-2 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrrr}-3&10&23&-13&14&-20&26&-14&-2\\& & -30& 21& -24& 30& -30& 12& \color{black}{6} \\ \hline &\color{blue}{10}&\color{blue}{-7}&\color{blue}{8}&\color{blue}{-10}&\color{blue}{10}&\color{blue}{-4}&\color{blue}{-2}&\color{orangered}{4} \end{array} $$The solution is:
$$ \frac{ 10x^{7}+23x^{6}-13x^{5}+14x^{4}-20x^{3}+26x^{2}-14x-2 }{ x+3 } = \color{blue}{10x^{6}-7x^{5}+8x^{4}-10x^{3}+10x^{2}-4x-2} ~+~ \frac{ \color{red}{ 4 } }{ x+3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-3}&10&23&-13&14&-20&26&-14&-2\\& & & & & & & & \\ \hline &&&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrrr}-3&\color{orangered}{ 10 }&23&-13&14&-20&26&-14&-2\\& & & & & & & & \\ \hline &\color{orangered}{10}&&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 10 } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-3}&10&23&-13&14&-20&26&-14&-2\\& & \color{blue}{-30} & & & & & & \\ \hline &\color{blue}{10}&&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 23 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrrrrrr}-3&10&\color{orangered}{ 23 }&-13&14&-20&26&-14&-2\\& & \color{orangered}{-30} & & & & & & \\ \hline &10&\color{orangered}{-7}&&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ 21 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-3}&10&23&-13&14&-20&26&-14&-2\\& & -30& \color{blue}{21} & & & & & \\ \hline &10&\color{blue}{-7}&&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 21 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrrrrr}-3&10&23&\color{orangered}{ -13 }&14&-20&26&-14&-2\\& & -30& \color{orangered}{21} & & & & & \\ \hline &10&-7&\color{orangered}{8}&&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 8 } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-3}&10&23&-13&14&-20&26&-14&-2\\& & -30& 21& \color{blue}{-24} & & & & \\ \hline &10&-7&\color{blue}{8}&&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrrrrrr}-3&10&23&-13&\color{orangered}{ 14 }&-20&26&-14&-2\\& & -30& 21& \color{orangered}{-24} & & & & \\ \hline &10&-7&8&\color{orangered}{-10}&&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-3}&10&23&-13&14&-20&26&-14&-2\\& & -30& 21& -24& \color{blue}{30} & & & \\ \hline &10&-7&8&\color{blue}{-10}&&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 30 } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrrrrrr}-3&10&23&-13&14&\color{orangered}{ -20 }&26&-14&-2\\& & -30& 21& -24& \color{orangered}{30} & & & \\ \hline &10&-7&8&-10&\color{orangered}{10}&&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 10 } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-3}&10&23&-13&14&-20&26&-14&-2\\& & -30& 21& -24& 30& \color{blue}{-30} & & \\ \hline &10&-7&8&-10&\color{blue}{10}&&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 26 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrrrrr}-3&10&23&-13&14&-20&\color{orangered}{ 26 }&-14&-2\\& & -30& 21& -24& 30& \color{orangered}{-30} & & \\ \hline &10&-7&8&-10&10&\color{orangered}{-4}&& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-3}&10&23&-13&14&-20&26&-14&-2\\& & -30& 21& -24& 30& -30& \color{blue}{12} & \\ \hline &10&-7&8&-10&10&\color{blue}{-4}&& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ 12 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrrrrr}-3&10&23&-13&14&-20&26&\color{orangered}{ -14 }&-2\\& & -30& 21& -24& 30& -30& \color{orangered}{12} & \\ \hline &10&-7&8&-10&10&-4&\color{orangered}{-2}& \end{array} $$Step 14 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-3}&10&23&-13&14&-20&26&-14&-2\\& & -30& 21& -24& 30& -30& 12& \color{blue}{6} \\ \hline &10&-7&8&-10&10&-4&\color{blue}{-2}& \end{array} $$Step 15 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 6 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrrrrr}-3&10&23&-13&14&-20&26&-14&\color{orangered}{ -2 }\\& & -30& 21& -24& 30& -30& 12& \color{orangered}{6} \\ \hline &\color{blue}{10}&\color{blue}{-7}&\color{blue}{8}&\color{blue}{-10}&\color{blue}{10}&\color{blue}{-4}&\color{blue}{-2}&\color{orangered}{4} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 10x^{6}-7x^{5}+8x^{4}-10x^{3}+10x^{2}-4x-2 } $ with a remainder of $ \color{red}{ 4 } $.