The textbook demonstrates the expansion of binomial squares using the identity (a + b)^2 = $a^{2}$ + 2ab + $b^{2}$ in Example 3. By substituting the first term of the binomial as 'a' and the second term as 'b', we can systematically expand each expression by squaring the first term, adding twice the product of both terms, and then adding the square of the second term.