x^2+(3a+1)x+2a^2-b^2

More generally:

.

If x^2+(3a+1)x+2a^2-b^2 is a perfect square, what is a,b?

for example:

$$a=-\dfrac 13 \text{ and } b = \dfrac {\sqrt{2}}{3} \quad \Rightarrow x^2$$