Possible value of quadratic expression coefficients

For $x^2 + bx + c = 0$ with roots $4a$ and $3a$:

• Sum of roots $= 4a + 3a = 7a = -b \implies b = -7a$
• Product of roots $= (4a)(3a) = 12a^2 = c$

Then: $b^2 + c = (-7a)^2 + 12a^2 = 49a^2 + 12a^2 = 61a^2$

Since $a$ is an integer, $b^2 + c$ must be 61 times a perfect square. Check 549: $\frac{549}{61} = 9 = 3^2 \implies a = 3$ Thus 549 is a possible value.