Divide using synthetic division $$ ( x^{2}+3x-43 ) \div ( x+8 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-8&1&3&-43\\& & -8& \color{black}{40} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{orangered}{-3} \end{array} $$The solution is:
$$ \frac{ x^{2}+3x-43 }{ x+8 } = \color{blue}{x-5} \color{red}{~-~} \frac{ \color{red}{ 3 } }{ x+8 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 8 = 0 $ ( $ x = \color{blue}{ -8 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-8}&1&3&-43\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-8&\color{orangered}{ 1 }&3&-43\\& & & \\ \hline &\color{orangered}{1}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ 1 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrr}\color{blue}{-8}&1&3&-43\\& & \color{blue}{-8} & \\ \hline &\color{blue}{1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrr}-8&1&\color{orangered}{ 3 }&-43\\& & \color{orangered}{-8} & \\ \hline &1&\color{orangered}{-5}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 40 } $.
$$ \begin{array}{c|rrr}\color{blue}{-8}&1&3&-43\\& & -8& \color{blue}{40} \\ \hline &1&\color{blue}{-5}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -43 } + \color{orangered}{ 40 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrr}-8&1&3&\color{orangered}{ -43 }\\& & -8& \color{orangered}{40} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{orangered}{-3} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x-5 } $ with a remainder of $ \color{red}{ -3 } $.