Divide using synthetic division $$ ( -41x^{2}-45x-36 ) \div ( x-7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}7&-41&-45&-36\\& & -287& \color{black}{-2324} \\ \hline &\color{blue}{-41}&\color{blue}{-332}&\color{orangered}{-2360} \end{array} $$The solution is:
$$ \frac{ -41x^{2}-45x-36 }{ x-7 } = \color{blue}{-41x-332} \color{red}{~-~} \frac{ \color{red}{ 2360 } }{ x-7 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -7 = 0 $ ( $ x = \color{blue}{ 7 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{7}&-41&-45&-36\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}7&\color{orangered}{ -41 }&-45&-36\\& & & \\ \hline &\color{orangered}{-41}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ \left( -41 \right) } = \color{blue}{ -287 } $.
$$ \begin{array}{c|rrr}\color{blue}{7}&-41&-45&-36\\& & \color{blue}{-287} & \\ \hline &\color{blue}{-41}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -45 } + \color{orangered}{ \left( -287 \right) } = \color{orangered}{ -332 } $
$$ \begin{array}{c|rrr}7&-41&\color{orangered}{ -45 }&-36\\& & \color{orangered}{-287} & \\ \hline &-41&\color{orangered}{-332}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ \left( -332 \right) } = \color{blue}{ -2324 } $.
$$ \begin{array}{c|rrr}\color{blue}{7}&-41&-45&-36\\& & -287& \color{blue}{-2324} \\ \hline &-41&\color{blue}{-332}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -36 } + \color{orangered}{ \left( -2324 \right) } = \color{orangered}{ -2360 } $
$$ \begin{array}{c|rrr}7&-41&-45&\color{orangered}{ -36 }\\& & -287& \color{orangered}{-2324} \\ \hline &\color{blue}{-41}&\color{blue}{-332}&\color{orangered}{-2360} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -41x-332 } $ with a remainder of $ \color{red}{ -2360 } $.