Divide using synthetic division $$ ( -3x^{4}+9x^{3}-12x^{2}+16x-15 ) \div ( -2x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-5&-3&9&-12&16&-15\\& & 15& -120& 660& \color{black}{-3380} \\ \hline &\color{blue}{-3}&\color{blue}{24}&\color{blue}{-132}&\color{blue}{676}&\color{orangered}{-3395} \end{array} $$The solution is:
$$ \frac{ -3x^{4}+9x^{3}-12x^{2}+16x-15 }{ x+5 } = \color{blue}{-3x^{3}+24x^{2}-132x+676} \color{red}{~-~} \frac{ \color{red}{ 3395 } }{ x+5 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&-3&9&-12&16&-15\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-5&\color{orangered}{ -3 }&9&-12&16&-15\\& & & & & \\ \hline &\color{orangered}{-3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&-3&9&-12&16&-15\\& & \color{blue}{15} & & & \\ \hline &\color{blue}{-3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 15 } = \color{orangered}{ 24 } $
$$ \begin{array}{c|rrrrr}-5&-3&\color{orangered}{ 9 }&-12&16&-15\\& & \color{orangered}{15} & & & \\ \hline &-3&\color{orangered}{24}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 24 } = \color{blue}{ -120 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&-3&9&-12&16&-15\\& & 15& \color{blue}{-120} & & \\ \hline &-3&\color{blue}{24}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ \left( -120 \right) } = \color{orangered}{ -132 } $
$$ \begin{array}{c|rrrrr}-5&-3&9&\color{orangered}{ -12 }&16&-15\\& & 15& \color{orangered}{-120} & & \\ \hline &-3&24&\color{orangered}{-132}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -132 \right) } = \color{blue}{ 660 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&-3&9&-12&16&-15\\& & 15& -120& \color{blue}{660} & \\ \hline &-3&24&\color{blue}{-132}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ 660 } = \color{orangered}{ 676 } $
$$ \begin{array}{c|rrrrr}-5&-3&9&-12&\color{orangered}{ 16 }&-15\\& & 15& -120& \color{orangered}{660} & \\ \hline &-3&24&-132&\color{orangered}{676}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 676 } = \color{blue}{ -3380 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&-3&9&-12&16&-15\\& & 15& -120& 660& \color{blue}{-3380} \\ \hline &-3&24&-132&\color{blue}{676}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ \left( -3380 \right) } = \color{orangered}{ -3395 } $
$$ \begin{array}{c|rrrrr}-5&-3&9&-12&16&\color{orangered}{ -15 }\\& & 15& -120& 660& \color{orangered}{-3380} \\ \hline &\color{blue}{-3}&\color{blue}{24}&\color{blue}{-132}&\color{blue}{676}&\color{orangered}{-3395} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -3x^{3}+24x^{2}-132x+676 } $ with a remainder of $ \color{red}{ -3395 } $.