Overview
The distributive property pairs each term of one factor with every term of the other. Combining the resulting like terms creates the middle coefficient. Squaring a binomial is a product of two identical groups, so it contains two cross-products.
Consider this example: Expand (x + 4)(x − 2). The four products are x², −2x, 4x, −8. Combine the middle terms to 2x, giving x² + 2x − 8. Dropping both cross-products gives x² − 8.
The method also handles a different case: Expand (2x − 3)(x + 5). The products give 2x² + 10x − 3x − 15, so the result is 2x² + 7x − 15. The coefficient 2 on x does not apply to the whole second factor independently.
For a further application, consider: The coefficient of x in (2x − a)(x + 3) is 1. What is the constant term of the expanded polynomial? Expansion gives 2x² + (6 − a)x − 3a. The middle coefficient condition 6 − a = 1 gives a = 5. Therefore the constant term is −3(5) = −15. Using a = 1 mistakes the specified coefficient for the parameter.
Squaring only the first and last terms omits the middle term. A difference of squares does lose its middle terms, but that cancellation depends on opposite signs. Expand systematically and then check the constant and leading coefficient. On a new square, expect a cross-term unless a coefficient or variable makes it zero.