Answer
a) $1+\frac{A}{t}+\frac{B}{t^2}+\frac{C}{t^3}+\frac{D}{1+t}+\frac{Ex+F}{t^2-t+1}$
b) $\frac{A}{x}+\frac{B}{x-1}+\frac{Cx+D}{x^2+1}+\frac{Ex+F}{(x^{2}+1)^2}$
Work Step by Step
a) $\frac{t^6+1}{t^6+t^3}$ = $1+\frac{1-t^3}{t^3(t^3+1)}$ =$1+\frac{1-t^3}{t^3(t+1)(t^2-t+1)}$ = $1+\frac{A}{t}+\frac{B}{t^2}+\frac{C}{t^3}+\frac{D}{1+t}+\frac{Ex+F}{t^2-t+1}$
b) $\frac{x^5+1}{(x^2-x)(x^{4}+2x^{2}+1)}$ = $\frac{x^5+1}{x(x-1)(x^{2}+1)^2}$ = $\frac{A}{x}+\frac{B}{x-1}+\frac{Cx+D}{x^2+1}+\frac{Ex+F}{(x^{2}+1)^2}$
