x1x2x3 + x1x2x4 + x1x3x4 + x2x3x4 <= C(x1^2 + x2^2 + x3^3 + x4^2)^{3/2}

I've having trouble showing for any four real numbers x1, x2, x3, x4,
there exists a real constant C such that

x1x2x3 + x1x2x4 + x1x3x4 + x2x3x4 <= C(x1^2 + x2^2 + x3^3 + x4^2)^{3/2}.

This is what I got so far: We can reduce it to the case of
considering the unit-sphere only. This is because the equation is
homogeneous. So, next I try to use lagrange multiplier to find the
maximum value of the left-hand side on the unit-sphere. However, the
algebra becomes unwielding and I cannot proceed.

Now this was an in-class exam question, and it should not have required
more than 15 minutes to solve. Am I missing a trick that I don't see?

I also tried using symmetry argument. The left-side side is symmetry
for every pair of coordinates. So, it's intuitive to guess that the
maximum occurs at x1=x2=x3=x4. However, I don't know how to justify
this rigorously.

Please help. Thanks.

-kira
.