What algebraic step is taken here?

Question

We have as part of a Laplace transform: $s^2F(s) - s - 1 + \frac{s^2F(s) - s - 1}{s^{\frac{1}{2}}} + F(s) = \frac{1}{s} + \frac{1}{s^2}\\ \implies F(s)(s^2+s^{\frac{3}{2}} + 1) = (\frac{1}{s} + \frac{1}{s^2})(s^2 + s^{\frac{3}{2}} + 1)\\ \implies F(s) = \frac{1}{s} + \frac{1}{s^2}$

where that first step is apparently seen through a direct insight, but I cannot figure out the leap taken.

Answer 1

You have

$$F(s)(s^2+s^{3/2}+1) - (s+1) -\frac{s+1}{s^{1/2}} =\frac{1}{s}+\frac{1}{s^2}.$$

Factor $s^2$ from the second term and $s^{3/2}$ from the third term

$$F(s)(s^2+s^{3/2}+1) - s^2\left(\frac{1}{s}+\frac{1}{s^2}\right) -s^{3/2}\left(\frac{1}{s}+\frac{1}{s^2}\right) =\frac{1}{s}+\frac{1}{s^2}.$$

Put the non-$F(s)$ stuff on the other side and factor out the $\left(\frac{1}{s}+\frac{1}{s^2}\right).$