If \(x, y\) are real numbers and equations \(x^2 - 12x + 35 = 0\) and \(x^2 + ax + 105 = 0\) have at least one common root, then the minimum possible value of \(y^2 + 4y - 5a\) is __________

The minimum is $106$. The first equation has roots $5$ and $7$, so substituting them in the second gives $a=-26$ and $a=-22$, respectively. Because $y^2+4y=(y+2)^2-4$, its minimum is $-4$, and choosing $a=-22$ gives $110-4=106$. Choosing $a=-26$ gives the higher minimum $126$.