Use special products to expand the following and simplify:

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

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The factor (a^(n+1) - 2)^2 is the square of a difference and we'll develop the binomial using the formula:

(x-y)^2 = x^2 - 2xy + y^2

Comparing, we'll get:

[a^(n+1) - 2]^2 = a^2(n+1) - 4a^(n+1) + 4

Now, we'll multiply the expansion above by 3a^n:

3a^n*[a^2(n+1) - 4a^(n+1) + 4] = 3a^[n+2(n+1)] - 12a^(n+n+1) + 12a^n

3a^n*[a^2(n+1) - 4a^(n+1) + 4] = 3a^(3n+2) - 12a^(2n+1) + 12a^n

**Using special products, we'll get the result 3a^(3n+2) - 12a^(2n+1) + 12a^n.**

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