<s>[INST] Polynomial solutions to $ax^4+bx^2+c^2 = y^2$, with $a,b,c$ as polynomials While investigating a system involving "equal sums of like powers", an elliptic curve popped up,
$$9 (1 + 4 n^4)^2 + 30 (4 + n^2 - 24 n^4 + 4 n^6) x^2 + 5 (32 - 40 n^2 + 53 n^4) x^4 = y^2$$
Some easy rational points are,
$$x = (0,\; 1,\; n)$$
Using the tangent-chord method, from the last two, we get,
$$\; x = \frac{6 (1 + 4 n^4)  (17 + 12 n^2)}{151 - 49 n^2 + 144 n^4 + 144 n^6}$$
$$x = \frac{\; 6 (1 + 4 n^4)  (3n + 23 n^3)}{9 + 9 n^2 - 79 n^4 + 121 n^6}$$
Question: But are there rational points where the numerator and denominator are polynomials of degree less than 6, preferably only quadratics?
 [/INST] Looking at a bunch of examples, it appears that your curve (which is isomorphic to an elliptic curve) has rank $2$ over $\mathbb{Q}(n)$. Using this, one can find some other rational points of degree $\leq 6$, but not where the numerator or denominator is a quadratic. In particular, there are solutions when
$$
  x = \frac{12 n^{4} + 3}{12 n^{3} \pm 23n^{2} + 17n \pm 3}.
$$
</s>