for his correct solution , which you can find below:
If $f(x) = x^4-8x^3+24x^2+bx+c$ has four real roots then its derivative must have three real roots. But $$f'(x) = 4x^3 - 24 x^2 + 48x + b = 4(x-2)^3 + b + 32,$$ and the function $(x-2)^3$ is strictly increasing except at the point $x=2$. So $f'(x)$ can only have three real roots if $b+32 = 0$. Therefore $b = -32$.