Given two generally non-square quartic polynomials that are to be simultaneously made squares for particular values of $x$,
$$c_1x^4+c_2x^3+c_3x^2+c_4x+c_5 = y_1^2$$
$$c_6x^4+c_7x^3+c_8x^2+c_9x+c_{10} = y_2^2$$
where the $c_i$ are rational constants.
Q: Is it proven that there can only be a finite number of rational $x,y_i$?
This problem sometimes arises in systems of equal sums of like powers. For example,
$$\begin{aligned}&(b - p)^k+(a + b + p)^k+(-a + b + p)^k+(2 + b - p)^k+(-2 + c)^k=\\ &(b + p)^k+(-a + b - p)^k+(a + b - p)^k+(-2 + b + p)^k+( 2 + c)^k\end{aligned}$$
which is good for $k = 1,3,5,7$, if,
$$2p+4 = a^2\tag{1}$$
$$p^2+3p-1 = 3b^2\tag{2}$$
$$10p^2+21p-1 = 3c^2\tag{3}$$
It is easy to satisfy the first condition, but the other two now become quartics to be made squares. There are an infinite number of non-trivial rational $p$ that simultaneously solve (1) and (2), OR (1) and (3), OR (2) and (3), but I would like to know if there are an infinite number of such $p$ that simultaneously satisfy all three.