I want to do a PFE of
$$ y(x) = \frac{1}{ x^2 + \sqrt{2}x +1} $$
when I try to expand the polynomial I end up with
$$ y(x) = \frac{1}{(x + \sqrt{\frac{1}{2}})( x + \sqrt{\frac{1}{2}}) + \frac{1}{2}} $$
I would know how to proceed if the $+\frac{1}{2}$ would not be there. That is what is confusing me. I tried
$$ y(x) = \frac{A}{x + \sqrt{\frac{1}{2}}} + \frac{B}{x + \sqrt{\frac{1}{2}} + \frac{1}{2}} $$
but it is not equivalent.
What you have is correct. If you are trying to integrate, the next step would be to substitute u = x + $1/\sqrt(2)$, then substitute again w = $u\sqrt(2)$ to get an expression of the form $\frac{1}{w^2 + 1}$ which integrates to arctangent w. If instead you are fine with complex numbers in your terms you can just use the quadratic formula as @Calvin Khor suggested. EDIT TO ADD: the Laplace transform of $ \sin kt = \frac {k}{s^2 + k^2}$ so adjust constants, using shift theorem as needed: the Laplace transform of $ e^{at} \sin kt = \frac{k}{(s-a)^2 + k^2} $