Divide using synthetic division $$ ( 14x^{3}-6x^{2}-22x-10 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-5&14&-6&-22&-10\\& & -70& 380& \color{black}{-1790} \\ \hline &\color{blue}{14}&\color{blue}{-76}&\color{blue}{358}&\color{orangered}{-1800} \end{array} $$The solution is:
$$ \frac{ 14x^{3}-6x^{2}-22x-10 }{ x+5 } = \color{blue}{14x^{2}-76x+358} \color{red}{~-~} \frac{ \color{red}{ 1800 } }{ x+5 } $$Explanation
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.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&14&-6&-22&-10\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-5&\color{orangered}{ 14 }&-6&-22&-10\\& & & & \\ \hline &\color{orangered}{14}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 14 } = \color{blue}{ -70 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&14&-6&-22&-10\\& & \color{blue}{-70} & & \\ \hline &\color{blue}{14}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -70 \right) } = \color{orangered}{ -76 } $
$$ \begin{array}{c|rrrr}-5&14&\color{orangered}{ -6 }&-22&-10\\& & \color{orangered}{-70} & & \\ \hline &14&\color{orangered}{-76}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -76 \right) } = \color{blue}{ 380 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&14&-6&-22&-10\\& & -70& \color{blue}{380} & \\ \hline &14&\color{blue}{-76}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -22 } + \color{orangered}{ 380 } = \color{orangered}{ 358 } $
$$ \begin{array}{c|rrrr}-5&14&-6&\color{orangered}{ -22 }&-10\\& & -70& \color{orangered}{380} & \\ \hline &14&-76&\color{orangered}{358}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 358 } = \color{blue}{ -1790 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&14&-6&-22&-10\\& & -70& 380& \color{blue}{-1790} \\ \hline &14&-76&\color{blue}{358}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ \left( -1790 \right) } = \color{orangered}{ -1800 } $
$$ \begin{array}{c|rrrr}-5&14&-6&-22&\color{orangered}{ -10 }\\& & -70& 380& \color{orangered}{-1790} \\ \hline &\color{blue}{14}&\color{blue}{-76}&\color{blue}{358}&\color{orangered}{-1800} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 14x^{2}-76x+358 } $ with a remainder of $ \color{red}{ -1800 } $.