Divide using synthetic division $$ ( 15x^{4}-14x^{3}-424x^{2}+350x+1225 ) \div ( x+5 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-5&15&-14&-424&350&1225\\& & -75& 445& -105& \color{black}{-1225} \\ \hline &\color{blue}{15}&\color{blue}{-89}&\color{blue}{21}&\color{blue}{245}&\color{orangered}{0} \end{array} $$The solution is:
$$ \dfrac{ 15x^{4}-14x^{3}-424x^{2}+350x+1225 }{ x+5 } = \color{blue}{15x^{3}-89x^{2}+21x+245} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&15&-14&-424&350&1225\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-5&\color{orangered}{ 15 }&-14&-424&350&1225\\& & & & & \\ \hline &\color{orangered}{15}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 15 } = \color{blue}{ -75 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&15&-14&-424&350&1225\\& & \color{blue}{-75} & & & \\ \hline &\color{blue}{15}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ \left( -75 \right) } = \color{orangered}{ -89 } $
$$ \begin{array}{c|rrrrr}-5&15&\color{orangered}{ -14 }&-424&350&1225\\& & \color{orangered}{-75} & & & \\ \hline &15&\color{orangered}{-89}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -89 \right) } = \color{blue}{ 445 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&15&-14&-424&350&1225\\& & -75& \color{blue}{445} & & \\ \hline &15&\color{blue}{-89}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -424 } + \color{orangered}{ 445 } = \color{orangered}{ 21 } $
$$ \begin{array}{c|rrrrr}-5&15&-14&\color{orangered}{ -424 }&350&1225\\& & -75& \color{orangered}{445} & & \\ \hline &15&-89&\color{orangered}{21}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 21 } = \color{blue}{ -105 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&15&-14&-424&350&1225\\& & -75& 445& \color{blue}{-105} & \\ \hline &15&-89&\color{blue}{21}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 350 } + \color{orangered}{ \left( -105 \right) } = \color{orangered}{ 245 } $
$$ \begin{array}{c|rrrrr}-5&15&-14&-424&\color{orangered}{ 350 }&1225\\& & -75& 445& \color{orangered}{-105} & \\ \hline &15&-89&21&\color{orangered}{245}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 245 } = \color{blue}{ -1225 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&15&-14&-424&350&1225\\& & -75& 445& -105& \color{blue}{-1225} \\ \hline &15&-89&21&\color{blue}{245}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 1225 } + \color{orangered}{ \left( -1225 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-5&15&-14&-424&350&\color{orangered}{ 1225 }\\& & -75& 445& -105& \color{orangered}{-1225} \\ \hline &\color{blue}{15}&\color{blue}{-89}&\color{blue}{21}&\color{blue}{245}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 15x^{3}-89x^{2}+21x+245 } $ with a remainder of $ \color{red}{ 0 } $.