To simplify the expression cos4a + sin2a cos2a, we can use the trigonometric identity cos2a = 1 - sin^2(a).

So, the expression becomes:

cos4a + sin2a (1 - sin^2(a))
= cos4a + sin2a - sin^3(a)

Now, we can further simplify by using the trigonometric identity cos(2x) = cos^2(x) - sin^2(x):

cos4a + sin2a - sin^3(a)
= (2cos^2(2a) - 1) + 2sin(a)cos(a) - sin^3(a)
= 2cos^2(2a) + 2sin(a)cos(a) - 1 - sin^3(a)