how to find rational numbers satisfying the binary quadratic equation $x^2+3xy+5y^2=4$ I am looking for a generalisation of the solution of $x,y$ wich are rational numbers,they could be infinite,how can i find such solutions,integer solutions are obvious
I have found that
$\Delta=-11,u=1,d=-11$
also
$(x+\frac{3+\sqrt{-11}y}{2})(x+\frac{3-\sqrt{-11}y}{2})=4$

This is a variation of Hecke's answer
$$(2x+3y)^2+11y^2=4^2$$
$$\left(\frac{2x}y+3\right)^2+11=\left(\frac4y\right)^2$$
$$11=\left(\frac4y\right)^2-\left(\frac{2x}y+3\right)^2$$
$$11=\left(\frac{4-2x}y-3\right)\left(\frac{4+2x}y+3\right)$$
$$\frac{4-2x}y-3=t, \frac{4+2x}y+3=\frac{11}t$$
$$\implies x = -\frac{2(t^2+6t-11)}{t^2+11},  y = \frac{8t}{t^2+11}, t\neq0$$

My comment contained an imprecision. This works for every equation that can be transformed into the form:
$$(ax+by)^2+cy^2=\left(\frac pq\right)^2$$
Where $a,b,c,p,q\in\mathbb Z, q\neq 0$
