Quadratic problem 1 by Dhaval Furia

Sum of roots of this quadratic equation $= -b$ . Product of roots of this quadratic equation $=c$ . But roots are given to be $4a$ and $3a$ . So,

$4a + 3a = -b$ and $4a\cdot 3a = c$

$\Rightarrow -b = 7a$ and $c = 12a^2$

$\Rightarrow b^2 = 49a^2$

$\Rightarrow b^2 + c = 49a^2 + 12a^2$

$\Rightarrow b^2 + c = 61a^2$

Since, $a$ is to be integer. Among the options given, only last option is a multiple of $61$ such that the quotient is a square. So

$61a^2 = 549$

$\Rightarrow a = 3$

Hence, among the given options, possible value of $b^2 + c$ is $549$ .