1. **Problem:** Factor the polynomial $4x^2 - 1$ using a special pattern. 2. **Formula and rules:** This is a difference of squares pattern: $$a^2 - b^2 = (a - b)(a + b)$$ where $a$ and $b$ are expressions. 3. **Identify $a$ and $b$:** Here, $4x^2 = (2x)^2$ and $1 = 1^2$, so $a = 2x$ and $b = 1$. 4. **Apply the formula:** $$4x^2 - 1 = (2x)^2 - 1^2 = (2x - 1)(2x + 1)$$ 5. **Intermediate step showing cancellation:** $$\cancel{4x^2} - \cancel{1} = (2x - 1)(2x + 1)$$ (illustrative cancellation of squares to factors) 6. **Final answer:** $$\boxed{(2x - 1)(2x + 1)}$$ This completes the factorization of the first polynomial.