Hi pals , possible to check where did i went wrong ?
2 / s(1 + s^2) = A/s + B/(1 + s^2)
then , 2 = A(1 + s^2) + Bs
when s = 0 , A = 2
when s = 1 and A = 2 , B = -2
however i could not get B = -2s
Because the denominator is an "irreducible" (unfactorable in terms of real numbers) quadratic. If x- a is a factor of the denominator, the partial fractions expansion will include $\displaystyle \frac{A}{x- a}$. If $\displaystyle x^2+ cx+ d= (x-a)^2+ b$, so that it is irreducible, then the partial fractions expansion will include $\displaystyle \frac{Ax+ B}{(x-a)^2+ b}$. If the denominator has $\displaystyle (x-a)^n$ as a factor, the partial fraction expansion will include $\displaystyle \frac{A_1}{x-a}+ \frac{A_2}{(x-a)^2}+ \cdot\cdot\cdot+ \frac{A_n}{(x-a)^n}$. Finally, if the denominator has $\displaystyle (x^2+ a)^n$ as a factor, the partial fraction expansion will include $\displaystyle \frac{A_1x+ B_1}{x^2+ a}+ \frac{A_2x+ B_2}{(x^2+a)^2}+$$\displaystyle \cdot\cdot\cdot+$$\displaystyle \frac{A_nx+ B_n}{(x^2+a)^n}$.
Since any polynomial, with real coefficients, can be factored (in the real numbers) to powers of linear or powers of quadratic terms, those are all the terms you need.