Divide using synthetic division $$ ( x^{3}-5x^{2}-9x+205 ) \div ( 4x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-5&1&-5&-9&205\\& & -5& 50& \color{black}{-205} \\ \hline &\color{blue}{1}&\color{blue}{-10}&\color{blue}{41}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{3}-5x^{2}-9x+205 }{ x+5 } = \color{blue}{x^{2}-10x+41} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&1&-5&-9&205\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-5&\color{orangered}{ 1 }&-5&-9&205\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 1 } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&1&-5&-9&205\\& & \color{blue}{-5} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrr}-5&1&\color{orangered}{ -5 }&-9&205\\& & \color{orangered}{-5} & & \\ \hline &1&\color{orangered}{-10}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 50 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&1&-5&-9&205\\& & -5& \color{blue}{50} & \\ \hline &1&\color{blue}{-10}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 50 } = \color{orangered}{ 41 } $
$$ \begin{array}{c|rrrr}-5&1&-5&\color{orangered}{ -9 }&205\\& & -5& \color{orangered}{50} & \\ \hline &1&-10&\color{orangered}{41}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 41 } = \color{blue}{ -205 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&1&-5&-9&205\\& & -5& 50& \color{blue}{-205} \\ \hline &1&-10&\color{blue}{41}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 205 } + \color{orangered}{ \left( -205 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-5&1&-5&-9&\color{orangered}{ 205 }\\& & -5& 50& \color{orangered}{-205} \\ \hline &\color{blue}{1}&\color{blue}{-10}&\color{blue}{41}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-10x+41 } $ with a remainder of $ \color{red}{ 0 } $.