Here, we have three equations for the product of two binomials, which are commonly used in algebra. The first formula is the general formula because the use of four distinct letters implies that all of the terms could be different. The second formula is for the square of a binomial, which is equivalent to the first formula with c = a and d = b. The third formula is called the difference of squares because of the right-hand side, and it is equivalent to the general formula with c = a and d = -b. The last two equations are just specializations of the first one.

Derivations

General Binomial Product

For the first equality, we apply the right-distributive property to get the right-hand side. Then we use the left-distributive property twice to get the second and final equality.

Square of a Binomial

To get the right-hand side of the first equality, we use the definition of an exponent. To get the second equality, we use the general formula that we derived above with c = a and d = b. For the third and final equality, we use the definition of exponents on the first and last terms and use commutativity and distributivity to combine the ab terms.

Difference of Squares

For the first equality, we use the general formula with c = a and d = -b to get the right-hand side. For the second inequality, we use the property that a product of an number and an additive inverse is the additive inverse of the product. For the third and final equality, we commutativity to say that ba = ab and the definition of an additive inverse to cancel the -ab and ab terms.