As Nasser mentioned, you can use Expand to fully expand your algebraic expression. If you want to know specifically the coefficients of powers of s then you can use CoefficientList as in the following:

In[1]:= CoefficientList[(4 + 2 s + s^2) (10 + 9 s + s^4), s]

Out[1]= {40, 56, 28, 9, 4, 2, 1}

If you just want the coefficient of a particular power of s then use Coefficient as in this:

In[2]:= Coefficient[(4 + 2 s + s^2) (10 + 9 s + s^4), s, 3]

Out[2]= 9

All of this assumes that you are working with polynomials rather than more complicated expressions...

What do you mean by expand with respect to s ? What is the output you are looking for and why Expand do not work for you?

Clear[s]
f = (4 + 2 s + s^2) (10 + 9 s + s^4);
Expand[f]
(*40 + 56 s + 28 s^2 + 9 s^3 + 4 s^4 + 2 s^5 + s^6*)