Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 1 / 8 / Solution

Multiply the partial fraction decomposition by $(x+1)(x+2{)}^2$ to obtain a polynomial identity:At $x=-1$, this gives $a=2$. Comparing the coefficients of $x^2$ then gives $a+b=1$, soThe remaining coefficients agree: $4a+3b-5=0$ and $4a+2b-5=1$. The original rational identity is understood away from its poles $-1$ and $-2$.