Divide using synthetic division $$ ( 7x^{2}+18x+45 ) \div ( 2x+5 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrr}-5&7&18&45\\& & -35& \color{black}{85} \\ \hline &\color{blue}{7}&\color{blue}{-17}&\color{orangered}{130} \end{array} $$The solution is:
$$ \dfrac{ 7x^{2}+18x+45 }{ x+5 } = \color{blue}{7x-17} ~+~ \dfrac{ \color{red}{ 130 } }{ x+5 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-5}&7&18&45\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-5&\color{orangered}{ 7 }&18&45\\& & & \\ \hline &\color{orangered}{7}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 7 } = \color{blue}{ -35 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&7&18&45\\& & \color{blue}{-35} & \\ \hline &\color{blue}{7}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ \left( -35 \right) } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrr}-5&7&\color{orangered}{ 18 }&45\\& & \color{orangered}{-35} & \\ \hline &7&\color{orangered}{-17}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ 85 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&7&18&45\\& & -35& \color{blue}{85} \\ \hline &7&\color{blue}{-17}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 45 } + \color{orangered}{ 85 } = \color{orangered}{ 130 } $
$$ \begin{array}{c|rrr}-5&7&18&\color{orangered}{ 45 }\\& & -35& \color{orangered}{85} \\ \hline &\color{blue}{7}&\color{blue}{-17}&\color{orangered}{130} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 7x-17 } $ with a remainder of $ \color{red}{ 130 } $.