$a=-54, b=217, c=-234$

$\mathrm{f}(x)=x^{4}+a x^{2}+b x+c$
$f(2)=0 \quad \Rightarrow 16+4 a+2 b+c=0 \Rightarrow 4 a+2 b+c=-16$…(1)
$\mathrm{f}(3)=12 \Rightarrow 81+9 a+3 b+c=12 \Rightarrow 9 a+3 b+c=-69 \ldots$..(2)
$\mathrm{f}(4)=26 \Rightarrow 256+16 a+4 b+c=26 \Rightarrow 16 a+4 b+c=-230 \ldots$..(3)
Solving (1), (2) and (3),
$a=-54, b=217, c=-234$