Simple algebra question - separating fractions

Question

How does the following come about? I'm completely lost. Can anyone help me fill in the steps in between?

$$ \frac{s+2}{s(s+1)} = \frac{2}{s} - \frac{1}{s+1} $$

I figured that $$ \frac{s+2}{s(s+1)} = \frac{s}{s(s+1)} + \frac{2}{s(s+1)} = \frac{1}{s+1} + (\text{not sure}) $$

Answer 1

HINT:

Using Partial Fraction Decomposition,

$$ \frac{s+2}{s(s+1)}=\frac As+\frac B{s+1}$$

$$ \implies s+2=A(s+1)+Bs$$ $$ \implies s+2=s(A+B)+A$$

$$ \text{Now, Compare the constants & the coefficients of $s$}$$

Answer 2

Well, you can note that $$s+2=(s+1)+1=(s+1)+(s+1-s)=2(s+1)-s,$$ so that $$\frac{s+2}{s(s+1)}=\frac{2(s+1)-s}{s(s+1)}=\frac{2(s+1)}{s(s+1)}-\frac s{s(s+1)}=\frac2s-\frac1{s+1}.$$