Divide using synthetic division $$ ( -10x^{4}-15x^{3}+9x^{2}-14x+17 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}3&-10&-15&9&-14&17\\& & -30& -135& -378& \color{black}{-1176} \\ \hline &\color{blue}{-10}&\color{blue}{-45}&\color{blue}{-126}&\color{blue}{-392}&\color{orangered}{-1159} \end{array} $$The solution is:
$$ \frac{ -10x^{4}-15x^{3}+9x^{2}-14x+17 }{ x-3 } = \color{blue}{-10x^{3}-45x^{2}-126x-392} \color{red}{~-~} \frac{ \color{red}{ 1159 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&-10&-15&9&-14&17\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}3&\color{orangered}{ -10 }&-15&9&-14&17\\& & & & & \\ \hline &\color{orangered}{-10}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&-10&-15&9&-14&17\\& & \color{blue}{-30} & & & \\ \hline &\color{blue}{-10}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ -45 } $
$$ \begin{array}{c|rrrrr}3&-10&\color{orangered}{ -15 }&9&-14&17\\& & \color{orangered}{-30} & & & \\ \hline &-10&\color{orangered}{-45}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -45 \right) } = \color{blue}{ -135 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&-10&-15&9&-14&17\\& & -30& \color{blue}{-135} & & \\ \hline &-10&\color{blue}{-45}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -135 \right) } = \color{orangered}{ -126 } $
$$ \begin{array}{c|rrrrr}3&-10&-15&\color{orangered}{ 9 }&-14&17\\& & -30& \color{orangered}{-135} & & \\ \hline &-10&-45&\color{orangered}{-126}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -126 \right) } = \color{blue}{ -378 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&-10&-15&9&-14&17\\& & -30& -135& \color{blue}{-378} & \\ \hline &-10&-45&\color{blue}{-126}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ \left( -378 \right) } = \color{orangered}{ -392 } $
$$ \begin{array}{c|rrrrr}3&-10&-15&9&\color{orangered}{ -14 }&17\\& & -30& -135& \color{orangered}{-378} & \\ \hline &-10&-45&-126&\color{orangered}{-392}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -392 \right) } = \color{blue}{ -1176 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&-10&-15&9&-14&17\\& & -30& -135& -378& \color{blue}{-1176} \\ \hline &-10&-45&-126&\color{blue}{-392}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ \left( -1176 \right) } = \color{orangered}{ -1159 } $
$$ \begin{array}{c|rrrrr}3&-10&-15&9&-14&\color{orangered}{ 17 }\\& & -30& -135& -378& \color{orangered}{-1176} \\ \hline &\color{blue}{-10}&\color{blue}{-45}&\color{blue}{-126}&\color{blue}{-392}&\color{orangered}{-1159} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -10x^{3}-45x^{2}-126x-392 } $ with a remainder of $ \color{red}{ -1159 } $.