Divide using synthetic division $$ ( 7x^{3}-30x^{2}+43x-10 ) \div ( -x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&7&-30&43&-10\\& & -14& 88& \color{black}{-262} \\ \hline &\color{blue}{7}&\color{blue}{-44}&\color{blue}{131}&\color{orangered}{-272} \end{array} $$The solution is:
$$ \frac{ 7x^{3}-30x^{2}+43x-10 }{ x+2 } = \color{blue}{7x^{2}-44x+131} \color{red}{~-~} \frac{ \color{red}{ 272 } }{ x+2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&7&-30&43&-10\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 7 }&-30&43&-10\\& & & & \\ \hline &\color{orangered}{7}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 7 } = \color{blue}{ -14 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&7&-30&43&-10\\& & \color{blue}{-14} & & \\ \hline &\color{blue}{7}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -30 } + \color{orangered}{ \left( -14 \right) } = \color{orangered}{ -44 } $
$$ \begin{array}{c|rrrr}-2&7&\color{orangered}{ -30 }&43&-10\\& & \color{orangered}{-14} & & \\ \hline &7&\color{orangered}{-44}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -44 \right) } = \color{blue}{ 88 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&7&-30&43&-10\\& & -14& \color{blue}{88} & \\ \hline &7&\color{blue}{-44}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 43 } + \color{orangered}{ 88 } = \color{orangered}{ 131 } $
$$ \begin{array}{c|rrrr}-2&7&-30&\color{orangered}{ 43 }&-10\\& & -14& \color{orangered}{88} & \\ \hline &7&-44&\color{orangered}{131}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 131 } = \color{blue}{ -262 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&7&-30&43&-10\\& & -14& 88& \color{blue}{-262} \\ \hline &7&-44&\color{blue}{131}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ \left( -262 \right) } = \color{orangered}{ -272 } $
$$ \begin{array}{c|rrrr}-2&7&-30&43&\color{orangered}{ -10 }\\& & -14& 88& \color{orangered}{-262} \\ \hline &\color{blue}{7}&\color{blue}{-44}&\color{blue}{131}&\color{orangered}{-272} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 7x^{2}-44x+131 } $ with a remainder of $ \color{red}{ -272 } $.