BG-MO 1982 Problem 5

Find all values of parameters $a,b$ for which the polynomial $$x^4+(2a+1)x^3+(a-1)^2x^2+bx+4$$ can be written as a product of two monic quadratic polynomials $\Phi(x)$ and $\Psi(x)$ , such that the equation $\Psi(x)=0$ has two distinct roots $\alpha,\beta$ which satisfy $\Phi(\alpha)=\beta$ and $\Phi(\beta)=\alpha$ .