Divide using synthetic division $$ ( -8x^{3}-x^{2}+62x-34 ) \div ( x-7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}7&-8&-1&62&-34\\& & -56& -399& \color{black}{-2359} \\ \hline &\color{blue}{-8}&\color{blue}{-57}&\color{blue}{-337}&\color{orangered}{-2393} \end{array} $$The solution is:
$$ \frac{ -8x^{3}-x^{2}+62x-34 }{ x-7 } = \color{blue}{-8x^{2}-57x-337} \color{red}{~-~} \frac{ \color{red}{ 2393 } }{ x-7 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{7}&-8&-1&62&-34\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}7&\color{orangered}{ -8 }&-1&62&-34\\& & & & \\ \hline &\color{orangered}{-8}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ -56 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&-8&-1&62&-34\\& & \color{blue}{-56} & & \\ \hline &\color{blue}{-8}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -56 \right) } = \color{orangered}{ -57 } $
$$ \begin{array}{c|rrrr}7&-8&\color{orangered}{ -1 }&62&-34\\& & \color{orangered}{-56} & & \\ \hline &-8&\color{orangered}{-57}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ \left( -57 \right) } = \color{blue}{ -399 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&-8&-1&62&-34\\& & -56& \color{blue}{-399} & \\ \hline &-8&\color{blue}{-57}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 62 } + \color{orangered}{ \left( -399 \right) } = \color{orangered}{ -337 } $
$$ \begin{array}{c|rrrr}7&-8&-1&\color{orangered}{ 62 }&-34\\& & -56& \color{orangered}{-399} & \\ \hline &-8&-57&\color{orangered}{-337}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ \left( -337 \right) } = \color{blue}{ -2359 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&-8&-1&62&-34\\& & -56& -399& \color{blue}{-2359} \\ \hline &-8&-57&\color{blue}{-337}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -34 } + \color{orangered}{ \left( -2359 \right) } = \color{orangered}{ -2393 } $
$$ \begin{array}{c|rrrr}7&-8&-1&62&\color{orangered}{ -34 }\\& & -56& -399& \color{orangered}{-2359} \\ \hline &\color{blue}{-8}&\color{blue}{-57}&\color{blue}{-337}&\color{orangered}{-2393} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -8x^{2}-57x-337 } $ with a remainder of $ \color{red}{ -2393 } $.