Divide using synthetic division $$ ( -7x^{2}+5x-1 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}3&-7&5&-1\\& & -21& \color{black}{-48} \\ \hline &\color{blue}{-7}&\color{blue}{-16}&\color{orangered}{-49} \end{array} $$The solution is:
$$ \frac{ -7x^{2}+5x-1 }{ x-3 } = \color{blue}{-7x-16} \color{red}{~-~} \frac{ \color{red}{ 49 } }{ x-3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -3 = 0 $ ( $ x = \color{blue}{ 3 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{3}&-7&5&-1\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}3&\color{orangered}{ -7 }&5&-1\\& & & \\ \hline &\color{orangered}{-7}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ -21 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&-7&5&-1\\& & \color{blue}{-21} & \\ \hline &\color{blue}{-7}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -21 \right) } = \color{orangered}{ -16 } $
$$ \begin{array}{c|rrr}3&-7&\color{orangered}{ 5 }&-1\\& & \color{orangered}{-21} & \\ \hline &-7&\color{orangered}{-16}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -16 \right) } = \color{blue}{ -48 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&-7&5&-1\\& & -21& \color{blue}{-48} \\ \hline &-7&\color{blue}{-16}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -48 \right) } = \color{orangered}{ -49 } $
$$ \begin{array}{c|rrr}3&-7&5&\color{orangered}{ -1 }\\& & -21& \color{orangered}{-48} \\ \hline &\color{blue}{-7}&\color{blue}{-16}&\color{orangered}{-49} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -7x-16 } $ with a remainder of $ \color{red}{ -49 } $.