Final answer:
The polynomial x⁴+2x²+1 is a perfect square trinomial which factors as (x²+1)². Since x²+1 does not factor over the real numbers, the fully factored form is (x²+1)² or (x + 1)²(x - 1)² if considering complex roots.
Step-by-step explanation:
The polynomial x⁴+2x²+1 can be factored by recognizing it as a perfect square trinomial. A perfect square trinomial is of the form (ax)² + 2abx + b², which can be factored into (ax+b)². In this case, the trinomial is (x²)² + 2(x²)(1) + (1)², which can be factored as (x²+1)². This expression further simplifies to (x + 1)(x + 1)(x - 1)(x - 1), or (x + 1)²(x - 1)², if we consider that x²+1 itself can be thought of as a difference of squares, x² - (-1)², although x²+1 does not factor over the real numbers since it involves the square root of a negative number.