kill(all)$

fpprintprec:4;

van=faceboook.com/LearningCalculus$

f(x):=horner(radcan(expand(
                      x^8 -  2*(1*sqrt(3)+1*sqrt(7))*x^7 +
      (11+4*sqrt(21))*x^6 -  2*(8*sqrt(3)+7*sqrt(7))*x^5 +
    2*(19+8*sqrt(21))*x^4 -  4*(8*sqrt(3)+7*sqrt(7))*x^3 +
    4*(11+4*sqrt(21))*x^2 - 16*(1*sqrt(3)+1*sqrt(7))*x   + 16)))$

u(x):=x^4-2*sqrt(3)*x^3+5*x^2-4*sqrt(3)*x+4;
v(x):=x^4-2*sqrt(7)*x^3+6*x^2-4*sqrt(7)*x+4;

g(x):=horner(radcan(expand(u(x)*v(x))))$ G(x)$

is(equal(f(x),g(x)));

us: solve(u(x),x); float(us);
vs: solve(v(x),x); float(vs);