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\(\boxed{1}\)**Question:
A quantum biologist is modeling the interaction between quantum states and genetic mutations. Given the expression $x^4 - 5x^2 + 4$, factor it completely.
The expression $x^4 - 5x^2 + 4$ is a quadratic in terms of $x^2$. Let $u = x^2$. The expression becomes $u^2 - 5u + 4$. We need to factor this quadratic.
Find two numbers that multiply to $4$ (the constant term) and add to $-5$ (the coefficient of the linear term).
The numbers $-4$ and $-1$ satisfy these conditions since $(-4) \times (-1) = 4$ and $(-4) + (-1) = -5$.
Thus, $u^2 - 5u + 4$ factors as $(u - 4)(u - 1)$.
Substitute back $u = x^2$:
Both $x^2 - 4$ and $x^2 - 1$ are differences of squares:
x^2 - 4 = (x - 2)(x + 2)
Therefore, the complete factorization is: