Divide using synthetic division $$ ( 2x^{3}+19x^{2}+36x-45 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-5&2&19&36&-45\\& & -10& -45& \color{black}{45} \\ \hline &\color{blue}{2}&\color{blue}{9}&\color{blue}{-9}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 2x^{3}+19x^{2}+36x-45 }{ x+5 } = \color{blue}{2x^{2}+9x-9} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&2&19&36&-45\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-5&\color{orangered}{ 2 }&19&36&-45\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 2 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&2&19&36&-45\\& & \color{blue}{-10} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 19 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrr}-5&2&\color{orangered}{ 19 }&36&-45\\& & \color{orangered}{-10} & & \\ \hline &2&\color{orangered}{9}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 9 } = \color{blue}{ -45 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&2&19&36&-45\\& & -10& \color{blue}{-45} & \\ \hline &2&\color{blue}{9}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 36 } + \color{orangered}{ \left( -45 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrr}-5&2&19&\color{orangered}{ 36 }&-45\\& & -10& \color{orangered}{-45} & \\ \hline &2&9&\color{orangered}{-9}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 45 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&2&19&36&-45\\& & -10& -45& \color{blue}{45} \\ \hline &2&9&\color{blue}{-9}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -45 } + \color{orangered}{ 45 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-5&2&19&36&\color{orangered}{ -45 }\\& & -10& -45& \color{orangered}{45} \\ \hline &\color{blue}{2}&\color{blue}{9}&\color{blue}{-9}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}+9x-9 } $ with a remainder of $ \color{red}{ 0 } $.