Expand the rational expression as
[tex]\dfrac ax+\dfrac{bx+c}{x^2+1}+\dfrac d{x-5}+\dfrac e{x+5}[/tex]
Then
[tex]2x^4+3x^3-8x^2-9x-10=a(x^2+1)(x^2-5)+(bx+c)x(x^2-5)+dx(x^2+1)(x+5)+ex(x^2+1)(x-5)[/tex]
Expand the right side, then set the coefficients of each term on both sides equal to each other and solve for the coefficients.
[tex]ax^4-4ax^2-5a+bx^4+cx^3-5bx^2-5cx+dx^4+5dx^3+dx^2+5dx+ex^4-5ex^3+ex^2-5ex[/tex]
[tex]=(a+b+d+e)x^4+(c+5d-5e)x^3+(-4a-5b+d+e)x^2+(-5c+5d-5e)x-5a[/tex]
[tex]\implies\begin{cases}a+b+d+e=2\\c+5d-5e=3\\-4a-5b+d+e=-8\\-5c+5d-5e=-9\\-5a=-10\end{cases}\implies\begin{cases}a=2\\b=0\\c=2\\d=\frac1{10},e=-\frac1{10}\end{cases}[/tex]
So the expansion is
[tex]\dfrac2x+\dfrac2{x^2+1}+\dfrac1{10x-50}-\dfrac1{10x+50}[/tex]
