The Square of a Binomial

(x+3)^2 is the expression this section has been warning about. The exponent does not distribute over the plus: the answer is not x^2 + 9. A power of a package means copies of the package: (x+3)(x+3).

Run the four products: x^2, 3x, 3x, 9. The cross products are twins, and they are exactly what x^2 + 9 forgets. Collected: x^2 + 6x + 9.

The general shape is worth keeping: (a+b)^2 = a^2 + 2ab + b^2. Square, double product, square. A quick check with numbers keeps everyone honest: (3+4)^2 = 49, while 3^2 + 4^2 = 25. The missing 24 is 2 * 3 * 4, the double product.