can this equation be factored further?

Question

y= x^4+2x^3+4x^2+8x+16

Not in the real number system. If you plot the function, you will see the minimum is at x=-1.1 (approx) and y is positive. At no x does the function equal zero, so there are no real roots, which means, no factors.
You can do some moving around..

x^4+2x^3+4x^2+8x+16
x^3(x + 2) + 4x(x + 2) + 16
(x+2)(x^3+ 4x) + 16
x(x +2)(x^2+4) + 16
I don't personally see that as progress.

Not having real roots doesn't mean it can't be factored: x^4+2x^2+1=(x^2+1)^2 has no real roots, but it does factor.
In theory every polynomial over R factors as polynomials of degree 1 or 2, but finding the factors can be time consuming in practice.

True.