Divide using synthetic division $$ ( x^{3}+5x^{2}-41x-45 ) \div ( x+9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-9&1&5&-41&-45\\& & -9& 36& \color{black}{45} \\ \hline &\color{blue}{1}&\color{blue}{-4}&\color{blue}{-5}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{3}+5x^{2}-41x-45 }{ x+9 } = \color{blue}{x^{2}-4x-5} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 9 = 0 $ ( $ x = \color{blue}{ -9 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&1&5&-41&-45\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-9&\color{orangered}{ 1 }&5&-41&-45\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 1 } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&1&5&-41&-45\\& & \color{blue}{-9} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrr}-9&1&\color{orangered}{ 5 }&-41&-45\\& & \color{orangered}{-9} & & \\ \hline &1&\color{orangered}{-4}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&1&5&-41&-45\\& & -9& \color{blue}{36} & \\ \hline &1&\color{blue}{-4}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -41 } + \color{orangered}{ 36 } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrr}-9&1&5&\color{orangered}{ -41 }&-45\\& & -9& \color{orangered}{36} & \\ \hline &1&-4&\color{orangered}{-5}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 45 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&1&5&-41&-45\\& & -9& 36& \color{blue}{45} \\ \hline &1&-4&\color{blue}{-5}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -45 } + \color{orangered}{ 45 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-9&1&5&-41&\color{orangered}{ -45 }\\& & -9& 36& \color{orangered}{45} \\ \hline &\color{blue}{1}&\color{blue}{-4}&\color{blue}{-5}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-4x-5 } $ with a remainder of $ \color{red}{ 0 } $.