The Square of a Binomial

$\left(x+3\right)^{2}$ is the square of a binomial. The exponent does not distribute over the plus sign, so the answer is not $x^{2}+9$. A power of a binomial means the binomial multiplied by itself: $\left(x+3\right)\left(x+3\right)$.

Write the four products: $x^{2}$, $3x$, $3x$ and $9$. The two cross products are equal, and they are the terms that $x^{2}+9$ leaves out. Collecting gives $x^{2}+6x+9$.

The square of a binomial is $\left(a+b\right)^{2} = a^{2}+2ab+b^{2}$. The middle term is the sum of the two equal cross products. Check with numbers: $\left(3+4\right)^{2} = 49$, but $3^{2}+4^{2} = 25$. The difference is $2\cdot 3\cdot 4 = 24$, which is the middle term that the second calculation leaves out.