The equation about x 4 x+2 x * a+a+ 1 = 0 has real roots, let y = 2x >；; 0,

It is equivalent to that y 2+ay+a+1= 0 has a positive root,

Equivalent to [-a+√ (A2-4A-4)]/2 >; 0,

Equivalent to √ (a 2-4a-4) > A, [note]

Equivalent to "a; =0, "or" a & gt=0, and a 2-4a-4 >; a^2'

The former a & lt=2-2√2, and the latter has no solution.

So a < =2-2√2 is demand.

[Note] To solve this unreasonable inequality, we need to square both sides, so we should discuss it in categories.