Partial Fractions- Is there a quicker way?

Question

So with the fraction

$$ (2s^2+5s+7)/(s+1)^3 $$

Is there a quicker way to solve this rather than equating the coefficients?

I cant use the 'cover-up' method because its just one fraction

This is what I got:

$$ A/(s+1) + B/(s+1)^2 + C/(s+1)^3 $$

Answer 1

hint: $2s^2 + 5s + 7 = 2(s+1)^2 + (s+1) + 4$

Answer 2

Using $2 s^{2} + 5 s + 7 = 2(s^{2} + 2 s + 1) + ( s + 1) + 4$ then \begin{align} \frac{2 s^{2} + 5 s + 7}{ (s+1)^{3} } = \frac{2}{s+1} + \frac{1}{(s+1)^{2}} + \frac{4}{(s+1)^{3}} \end{align}

Answer 3

Let $t = s+1$. Then $\frac{2s^2+5s+7}{(s+1)^3} =\frac{2(t-1)^2+5(t-1)+7}{t^3} =\frac{2t^2-4t+2+5t-5+7}{t^3} =\frac{2t^2+t+4}{t^3} $.