Expand algebraic expressions with the perfect square and difference of two squares identities

Follow this one with me, a step at a time. Try to guess each step first.

Worked example 1

Expand (2x + 9y)².

Remember: A perfect square has three terms: the first squared, twice the product, the last squared.

Use (a + b)² ≡ a² + 2ab + b² with a = 2x and b = 9y.
That is (2x)² + 2(2x)(9y) + (9y)².
The answer is 4x² + 36xy + 81y².

Common mistakes

Squared the letter and not its coefficient. The whole term is squared.
The middle term should be twice the product of the two terms.
The middle term takes the sign in the bracket, so (a − b)² has −2ab.