Divide using synthetic division $$ ( x^{4}+8x^{3}+24x^{2}+25x-14 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-3&1&8&24&25&-14\\& & -3& -15& -27& \color{black}{6} \\ \hline &\color{blue}{1}&\color{blue}{5}&\color{blue}{9}&\color{blue}{-2}&\color{orangered}{-8} \end{array} $$The solution is:
$$ \frac{ x^{4}+8x^{3}+24x^{2}+25x-14 }{ x+3 } = \color{blue}{x^{3}+5x^{2}+9x-2} \color{red}{~-~} \frac{ \color{red}{ 8 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&8&24&25&-14\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 1 }&8&24&25&-14\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 1 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&8&24&25&-14\\& & \color{blue}{-3} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrrr}-3&1&\color{orangered}{ 8 }&24&25&-14\\& & \color{orangered}{-3} & & & \\ \hline &1&\color{orangered}{5}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 5 } = \color{blue}{ -15 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&8&24&25&-14\\& & -3& \color{blue}{-15} & & \\ \hline &1&\color{blue}{5}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrrr}-3&1&8&\color{orangered}{ 24 }&25&-14\\& & -3& \color{orangered}{-15} & & \\ \hline &1&5&\color{orangered}{9}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 9 } = \color{blue}{ -27 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&8&24&25&-14\\& & -3& -15& \color{blue}{-27} & \\ \hline &1&5&\color{blue}{9}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 25 } + \color{orangered}{ \left( -27 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}-3&1&8&24&\color{orangered}{ 25 }&-14\\& & -3& -15& \color{orangered}{-27} & \\ \hline &1&5&9&\color{orangered}{-2}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&8&24&25&-14\\& & -3& -15& -27& \color{blue}{6} \\ \hline &1&5&9&\color{blue}{-2}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ 6 } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrr}-3&1&8&24&25&\color{orangered}{ -14 }\\& & -3& -15& -27& \color{orangered}{6} \\ \hline &\color{blue}{1}&\color{blue}{5}&\color{blue}{9}&\color{blue}{-2}&\color{orangered}{-8} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}+5x^{2}+9x-2 } $ with a remainder of $ \color{red}{ -8 } $.