Inspired by incomplete Muirhead

We attempt use lagrange multipliers to solve this problem. Let $f(a,b,c,d) = a^2bc+b^2cd$ and $g(a,b,c,d) = a+b+c+d-5$ . We then need to find a point in 4-dimensional space with positive coordinates with sum 5 in which a stationary point is obtained

$f(a,b,c,d))-\lambda(g(a,b,c,d))= a^2bc+b^2cd- \lambda(a+b+c+d-5)=0$ . Differentiating both sides with respect to $a,b,c,d$ respectively gives us $\lambda = 2abc=a^2c+2bcd=a^2b+b^2d=b^2c$

Solving for the ratios of $a,b,c,d$ to each other gives us $b=2a, d=\frac{3a}{4}, c=\frac{5a}{4}$ . This gives us $(a,b,c,d) = (1,2,\frac{5}{4},\frac{3}{4})$ . Substituting this into the original equation, we get that the value at the stationary point is $\frac{25}{2^2}$ .

But, since $(0.1, 2.9, 1,1)$ yields a higher value than the value at the turning point, we see that the turning point cannot be a maxima. This implies that there is no maxima for this range of values. Indeed, the maxima can only be obtained on the set of nonnegative real numbers.

So, our answer is $\boxed{490023}$