Divide using synthetic division $$ ( -62x^{4}+x^{2}-15 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-4&-62&0&1&0&-15\\& & 248& -992& 3964& \color{black}{-15856} \\ \hline &\color{blue}{-62}&\color{blue}{248}&\color{blue}{-991}&\color{blue}{3964}&\color{orangered}{-15871} \end{array} $$The solution is:
$$ \frac{ -62x^{4}+x^{2}-15 }{ x+4 } = \color{blue}{-62x^{3}+248x^{2}-991x+3964} \color{red}{~-~} \frac{ \color{red}{ 15871 } }{ x+4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-62&0&1&0&-15\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-4&\color{orangered}{ -62 }&0&1&0&-15\\& & & & & \\ \hline &\color{orangered}{-62}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -62 \right) } = \color{blue}{ 248 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-62&0&1&0&-15\\& & \color{blue}{248} & & & \\ \hline &\color{blue}{-62}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 248 } = \color{orangered}{ 248 } $
$$ \begin{array}{c|rrrrr}-4&-62&\color{orangered}{ 0 }&1&0&-15\\& & \color{orangered}{248} & & & \\ \hline &-62&\color{orangered}{248}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 248 } = \color{blue}{ -992 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-62&0&1&0&-15\\& & 248& \color{blue}{-992} & & \\ \hline &-62&\color{blue}{248}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -992 \right) } = \color{orangered}{ -991 } $
$$ \begin{array}{c|rrrrr}-4&-62&0&\color{orangered}{ 1 }&0&-15\\& & 248& \color{orangered}{-992} & & \\ \hline &-62&248&\color{orangered}{-991}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -991 \right) } = \color{blue}{ 3964 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-62&0&1&0&-15\\& & 248& -992& \color{blue}{3964} & \\ \hline &-62&248&\color{blue}{-991}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 3964 } = \color{orangered}{ 3964 } $
$$ \begin{array}{c|rrrrr}-4&-62&0&1&\color{orangered}{ 0 }&-15\\& & 248& -992& \color{orangered}{3964} & \\ \hline &-62&248&-991&\color{orangered}{3964}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 3964 } = \color{blue}{ -15856 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-62&0&1&0&-15\\& & 248& -992& 3964& \color{blue}{-15856} \\ \hline &-62&248&-991&\color{blue}{3964}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ \left( -15856 \right) } = \color{orangered}{ -15871 } $
$$ \begin{array}{c|rrrrr}-4&-62&0&1&0&\color{orangered}{ -15 }\\& & 248& -992& 3964& \color{orangered}{-15856} \\ \hline &\color{blue}{-62}&\color{blue}{248}&\color{blue}{-991}&\color{blue}{3964}&\color{orangered}{-15871} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -62x^{3}+248x^{2}-991x+3964 } $ with a remainder of $ \color{red}{ -15871 } $.