To find the coefficient of x^4 in the expansion of (1-x+2x^2)^12, we will use the binomial theorem. The binomial theorem states that for any real numbers a and b, and any positive integer n, the expansion of (a + b)^n can be written as:

(a + b)^n = C(n, 0)a^n + C(n, 1)a^(n-1)b + C(n, 2)a^(n-2)b^2 + ... + C(n, n-1)ab^(n-1) + C(n, n)b^n

Where C(n, k) represents the binomial coefficient, given by:

C(n, k) = n! / (k!(n-k)!)

Now, let's apply the binomial theorem to our expression.



In our expression, a = 1, b = -x + 2x^2, and n = 12. We want to find the coefficient of x^4, so we need to determine the term in the expansion of (1-x+2x^2)^12 that contains x^4.

The general term in the expansion is given by:

T(k) = C(12, k) * (1)^(12-k) * (-x + 2x^2)^k

For the coefficient of x^4, we need to find the value of k that satisfies the condition (12-k) + 2k = 4.

Simplifying the equation, we get:
12 - k + 2k = 4
12 + k = 4
k = 4 - 12
k = -8

Since k cannot be negative, there is no term in the expansion of (1-x+2x^2)^12 that contains x^4. Therefore, the coefficient of x^4 is 0.

In other words, when expanding (1-x+2x^2)^12, there is no term that contains x^4.

Therefore, the coefficient of x^4 in the expansion of (1-x+2x^2)^12 is 0.