We want to calculate the following division
[tex]\frac{8x^4-15x^3-2x^2+x-2}{x-2}[/tex]
Using the long division method, we start by dividing the leading term of the dividend by the leading term of the divisor
[tex]\frac{8x^4}{x}=8x^3[/tex]
Then, we multiply it by the divisor
[tex]8x^3(x-2)=8x^4-16x^3[/tex]
then, subtract the dividend from the obtained result
[tex](8x^4-15x^3-2x^2+x-2)-(8x^4-16x^3)=x^3-2x^2+x-2[/tex]
Then, our division can be rewritten as
[tex]\frac{8x^4-15x^3-2x^2+x-2}{x-2}=8x^{3^{}}+\frac{x^3-2x^2+x-2}{x-2}[/tex]
Repeating the whole process for the remaining division, we have
[tex]\begin{gathered} \frac{x^3}{x}=x^2 \\ x^2(x-2)=x^3-2x^2 \\ (x^3-2x^2+x-2)-(x^3-2x^2)=x-2 \\ \Rightarrow8x^{3^{}}+\frac{x^3-2x^2+x-2}{x-2}=8x^{3^{}}+x^2+\frac{x-2}{x-2} \end{gathered}[/tex]
Repeating the whole process again, we have our result
[tex]\frac{8x^4-15x^3-2x^2+x-2}{x-2}=8x^{3^{}}+x^2+1+\frac{0}{x-2}[/tex]
