First, we will expand the terms in the expression:

(a+1)^2 = a^2 + 2a + 1

2(a+1) = 2a + 2

-3(a-1)(a+1) = -3(a^2 - a + a - 1) = -3(a^2 - 1)

Now, we can substitute those expanded terms back into the original expression:

(a^2 + 2a + 1) + (2a + 2) - (-3(a^2 - 1))

Simplify by combining like terms:

a^2 + 2a + 1 + 2a + 2 + 3a^2 + 3

Now, combine like terms:

a^2 + 3a^2 + 2a + 2a + 1 + 2 + 3

Finally, combine all the terms together:

4a^2 + 4a + 6

So, the simplified form of the expression (a+1)^2 + 2(a+1) - 3(a-1)(a+1) is 4a^2 + 4a + 6.