y$ \;=\; \dfrac{x^2+4}{(x-1)^2(2x+3)} \; = \; \dfrac{x^2+4}{(x^2-2x+1)(2x+3)} \\ \; \\ \; \\ y \;=\; \dfrac{(x^2-2x+1)+(2x+3)}{(x^2-2x+1)(2x+3)} \; = \; \dfrac{1}{2x+3} \; + \; \dfrac{1}{(x-1)^2} \\ \; \\ \; \\ Diff \; w.r.t \; x,, \\ \; \\ \therefore y_1 \; = \; \dfrac{-2}{(2x+3)^2} \; - \; \dfrac{2}{(x-1)^3} \\ \; \\ \; \\ Diff \; again \; w.r.t \; x, \\ \; \\ \therefore y_2 \; = \; \dfrac{(-2)(-2)(2)}{(2x+3)^3} \; + \; \dfrac{6}{(x-1)^4} \; = \; \dfrac{8}{(2x+3)^3} \; + \; \dfrac{6}{(x-1)^4} \; = \; \dfrac{(-2)^2 2!}{(2x+3)^3} \; + \; \dfrac{3!}{(x-1)^4} \\ \; \\ \; \\ Similarly, \\ \; \\ y_3 \; = \; \dfrac{-48}{(2x+3)^4} \; - \; \dfrac{24}{(x-1)^5} \; = \; \dfrac{(-2)^3 3!}{(2x+3)^{3+1}} \; + \; \dfrac{4!(-1)^3}{(x-1)^{3+2}} \\ \; \\ \; \\ \; \\ Diff \; w.r.t \; x, \; 'n' \; times \; \; \\ \; \\ \; \\ y_n \; = \; \dfrac{(-2)^n n!}{(2x+3)^{n+1}} \; + \; \dfrac{(n+1)!(-1)^n}{(x-1)^{n+2}} $