The textbook context introduces algebraic identities such as $(x+a)(x+b) = x^2 + (a+b)x + ab$ and suggests evaluating squares like $35^2$ using suitable identities. By extending these patterns, we can use the standard identities for squaring and product of terms: $(a+b)^2 = a^2 + 2ab + b^2$, $(a-b)^2 = a^2 - 2ab + b^2$, and $(x+a)(x+b) = x^2 + (a+b)x + ab$ to find the products without direct multiplication.