Divide using synthetic division $$ ( -150x^{2}+1000x+480000 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}5&-150&1000&480000\\& & -750& \color{black}{1250} \\ \hline &\color{blue}{-150}&\color{blue}{250}&\color{orangered}{481250} \end{array} $$The solution is:
$$ \frac{ -150x^{2}+1000x+480000 }{ x-5 } = \color{blue}{-150x+250} ~+~ \frac{ \color{red}{ 481250 } }{ x-5 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{5}&-150&1000&480000\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}5&\color{orangered}{ -150 }&1000&480000\\& & & \\ \hline &\color{orangered}{-150}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -150 \right) } = \color{blue}{ -750 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&-150&1000&480000\\& & \color{blue}{-750} & \\ \hline &\color{blue}{-150}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1000 } + \color{orangered}{ \left( -750 \right) } = \color{orangered}{ 250 } $
$$ \begin{array}{c|rrr}5&-150&\color{orangered}{ 1000 }&480000\\& & \color{orangered}{-750} & \\ \hline &-150&\color{orangered}{250}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 250 } = \color{blue}{ 1250 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&-150&1000&480000\\& & -750& \color{blue}{1250} \\ \hline &-150&\color{blue}{250}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 480000 } + \color{orangered}{ 1250 } = \color{orangered}{ 481250 } $
$$ \begin{array}{c|rrr}5&-150&1000&\color{orangered}{ 480000 }\\& & -750& \color{orangered}{1250} \\ \hline &\color{blue}{-150}&\color{blue}{250}&\color{orangered}{481250} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -150x+250 } $ with a remainder of $ \color{red}{ 481250 } $.