Description: Given the following parametric cubic polynomials ${E}^{3} - 15\, {\beta}_{\pm}\, {E}^{2} - 3 \left({71\, {\beta}_{\pm}^{2} + 352\, {\beta}_{\mp}}\right) E + 135\, {\beta}_{\pm} \left({5\, {\beta}_{\pm}^{2} - 32\, {\beta}_{\mp}}\right)$ where ${\beta}_{\pm} = 1 \pm \eta$, how do I find all possible rational values of $\eta$ such that this polynomial is reducible.

Background: I have found 61 rational values of $\eta$ that factor these polynomials into a rational linear $E - p/q$ for integer $p$ and $q$ relatively prime and $q \ne 0$ times an irreducible quadratic of the form ${E}^{2} + a\, E + b$. I do not know if there are an infinite number of such values and if not do I have the complete set.