Question:

Step 1: Identify the polynomial and its relationship to the product. We are given the polynomial $p(x) = x^4 + (2 - \sqrt{3})x^2 + (2 + \sqrt{3})$ with roots $\alpha_1, \alpha_2, \alpha_3, \alpha_4$. Since these are the roots, we can write: $$p(x) = (x - \alpha_1)(x - \alpha_2)(x - \alpha_3)(x - \alpha_4)$$ Step 2: Express the desired product in terms of the polynomial. We need to find $(1 - \alpha_1)(1 - \alpha_2)(1 - \alpha_3)(1 - \alpha_4)$. Notice that: $$(1 - \alpha_1)(1 - \alpha_2)(1 - \alpha_3)(1 - \alpha_4) = (-({\alpha_1 - 1}))(-({\alpha_2 - 1}))(-({\alpha_3 - 1}))(-({\alpha_4 - 1}))$$ Since there are 4 negative signs (an even number), they cancel out: $$(1 - \alpha_1)(1 - \alpha_2)(1 - \alpha_3)(1 - \alpha_4) = (\alpha_1 - 1)(\alpha_2 - 1)(\alpha_3 - 1)(\alpha_4 - 1)$$ This is equivalent to evaluating $p(1)$ because: $$p(1) = (1 - \alpha_1)(1 - \alpha_2)(1 - \alpha_3)(1 - \alpha_4)$$ Step 3: Evaluate the polynomial at $x = 1$. Substitute $x = 1$ into $p(x)$: $$p(1) = (1)^4 + (2 - \sqrt{3})(1)^2 + (2 + \sqrt{3})$$ $$p(1) = 1 + (2 - \sqrt{3}) + (2 + \sqrt{3})$$ Step 4: Simplify the expression. Combine like terms: $$p(1) = 1 + 2 - \sqrt{3} + 2 + \sqrt{3}$$ $$p(1) = 1 + 2 + 2 + (-\sqrt{3} + \sqrt{3})$$ $$p(1) = 5 + 0 = 5$$ Step 5: State the final answer. Therefore, $(1 - \alpha_1)(1 - \alpha_2)(1 - \alpha_3)(1 - \alpha_4) = 5$. The answer is **Option 4: 5**.