1. The problem is to square the expression $x^2 - 81$. 2. The formula for squaring a binomial $(a - b)^2$ is: $$ (a - b)^2 = a^2 - 2ab + b^2 $$ 3. Here, $a = x^2$ and $b = 81$. 4. Applying the formula: $$ (x^2 - 81)^2 = (x^2)^2 - 2 \cdot x^2 \cdot 81 + 81^2 $$ 5. Simplify each term: $$ (x^2)^2 = x^{4} $$ $$ -2 \cdot x^2 \cdot 81 = -162x^2 $$ $$ 81^2 = 6561 $$ 6. So the squared expression is: $$ x^{4} - 162x^{2} + 6561 $$ This is the expanded form of the square of $x^2 - 81$.