To calculate the square root of a polynomial, one must find a polynomial p(x) such that [p(x)]² equals the given polynomial q(x). For the polynomial 4x^4 + 8x^3 + 8x^2 + 4x + 1, it is not possible to find such a p(x) because q must have roots with even multiplicities, which this polynomial does not possess. Attempting to express p as ax² + bx + c and squaring it leads to a system of equations that has no solution. Therefore, finding the square root of this specific polynomial is not feasible. Understanding the conditions for polynomial roots is crucial in these calculations.