\[ \text{Find } 2011^2 - 2010^2 \text{?} \]

To solve \( 2011^2 - 2010^2 \), we can use the difference of squares formula, which states that \( a^2 - b^2 = (a - b)(a + b) \). Here, let \( a = 2011 \) and \( b = 2010 \). So, substituting into the formula: \[ 2011^2 - 2010^2 = (2011 - 2010)(2011 + 2010) \] Calculating each part: \[ 2011 - 2010 = 1 \] \[ 2011 + 2010 = 4021 \] Now, plug these values back into the equation: \[ 2011^2 - 2010^2 = 1 \times 4021 = 4021 \] Thus, the answer is \( \boxed{4021} \). Now, here's some additional insightful and engaging context: Did you know that the difference of squares is a classic algebraic identity that dates back centuries? It's been a fundamental part of mathematics since ancient civilizations tackled geometric problems! This method not only makes calculations simpler but also serves as a bridge to understanding polynomials and quadratic equations. If you're ever in doubt about using the difference of squares formula, remember it can be a lifesaver! One common mistake is forgetting to properly identify \( a \) and \( b \). Always double-check your terms; it’s the key to tackling these problems efficiently and avoiding errors!