(1+sqrt(2)+sqrt(3)+sqrt(6))^2=1+sqrt(2)+sqrt(3)+sqrt(6)+sqrt(2)+2+sqrt(6)+sqrt(12)+sqrt(3)+sqrt(6)+3+sqrt(18)+sqrt(6)+sqrt(12)+sqrt(18)+6=12+2*sqrt(2)+2*sqrt(3)+4*sqrt(6)+2*sqrt(12)+2*sqrt(18)=12+2*sqrt(2)+2*sqrt(3)+4*sqrt(6)+4*sqrt(3)+6*sqrt(2)=12+8*sqrt(2)+6*sqrt(3)+4*sqrt(6)

Now, subtract 21 to get -9+8*sqrt(2)+6*sqrt(3)+4*sqrt(6). Since 9*8*sqrt(2)=72*sqrt(2)=24*sqrt(18)=6*sqrt(3)*4*sqrt(6), then the sqrt(2) terms will cancel when we square it.

(-9+8*sqrt(2)+6*sqrt(3)+4*sqrt(6))^2=81-72*sqrt(2)-54*sqrt(3)-36*sqrt(6)-72*sqrt(2)+128+48*sqrt(6)+32*sqrt(12)-54*sqrt(3)+48*sqrt(6)+108+24*sqrt(18)-36*sqrt(6)+32*sqrt(12)+24*sqrt(18)+96=413-144*sqrt(2)-108*sqrt(3)+24*sqrt(6)+64*sqrt(12)+48*sqrt(18)=413-144*sqrt(2)-108*sqrt(3)+24*sqrt(6)+128*sqrt(3)+144*sqrt(2)=413+20*sqrt(3)+24*sqrt(6)

Now, remove the 413 to get 20*sqrt(3)+24*sqrt(6). Then, square it.

(20*sqrt(3)+24*sqrt(6))^2=1200+480*sqrt(18)+480*sqrt(18)+3456=4656+960*sqrt(18)=4656+sqrt(16588800)

Therefore, 1+sqrt(2)+sqrt(3)+sqrt(6)=sqrt(12+8*sqrt(2)+6*sqrt(3)+4*sqrt(6))=sqrt(21-9+8*sqrt(2)+6*sqrt(3)+4*sqrt(6))=sqrt(21+sqrt(413+20*sqrt(3)+24*sqrt(6)))=sqrt(21+sqrt(413+sqrt(4656+sqrt(16588800)))).

a=21

b=413

c=4656

d=16588800