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Factor x^(4)+12x^(2)+36 completely. All factors in your answer should have integer coefficients.

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To factor x^(4)+12x^(2)+36, we familiarize ourselves with patterns that lead to perfect squares. Considering that a quadratic equation like y = ax^2 + bx + c can be factored as a perfect square only if b^2 = 4ac, we note that the given equation is in the form of u^2 + 2vu + v^2, where u = x^2 and v = 6.

1. The given polynomial is x^4 + 12x^2 + 36.

2. From this, we can observe that x^4 is a perfect fourth power, 12*x^2 is twice the product of x^2 and 6, and 36 is a perfect square of 6.

3. Therefore, our polynomial satisfies the identity a^2+2ab+b^2 = (a+b)^2. Here, a = x^2 and b = 6.

4. Therefore, we can factorize x^4 + 12x^2 + 36 as (x^2 + 6)^2.

Our factorized polynomial is (x^2 + 6)^2.

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