This expression can be simplified using trigonometric identities.

First, we can use the trigonometric identity sin^2(alpha) + cos^2(alpha) = 1.

So, sin^4(alpha) = (sin^2(alpha))^2 and cos^4(alpha) = (cos^2(alpha))^2.

Therefore, the expression becomes:

(sin^2(alpha))^2 + (cos^2(alpha))^2 + 2sin(alpha)cos^2(alpha)

Now, using the identity sin^2(alpha) + cos^2(alpha) = 1, we can rewrite the expression as:

1 + 1 + 2sin(alpha)cos^2(alpha)

Simplifying further, we get:

2 + 2sin(alpha)cos^2(alpha)

Which can also be written as:

2 + 2sin(alpha)cos(alpha)

Therefore, the simplified expression is:

2 + 2sin(alpha)cos(alpha)