Expanding the Expression: (x+2)(x^2  2x + 4)
This expression represents the multiplication of a binomial and a trinomial. To simplify it, we can use the distributive property. This means we multiply each term of the binomial (x + 2) by each term of the trinomial (x^2  2x + 4).
Here's how it works:

Multiply x by each term of the trinomial:
 x * x^2 = x^3
 x * (2x) = 2x^2
 x * 4 = 4x

Multiply 2 by each term of the trinomial:
 2 * x^2 = 2x^2
 2 * (2x) = 4x
 2 * 4 = 8

Combine all the terms:
 x^3  2x^2 + 4x + 2x^2  4x + 8

Simplify by combining like terms:
 x^3 + 8
Therefore, the expanded form of (x+2)(x^2  2x + 4) is x^3 + 8.
Interesting Observation:
This expression is a special case of the sum of cubes factorization. Notice that the trinomial (x^2  2x + 4) is a result of squaring the first term of the binomial (x), changing the sign of the second term (2x), and squaring the second term (2). This pattern always leads to a simplified expression with just two terms:
(a + b)(a^2  ab + b^2) = a^3 + b^3
In our case, a = x and b = 2.