How do you find the coefficient of $x$ in $(x + 1)^2$? I want to learn how can I find out the coefficient of the variable $x$ in the expression $(x + 1)^2$. It is a case of a perfect square expansion.

This is the distributive property of the multiplication: $(a+b)c=ac+bc$. 
What is $(x+a)^2$? Well,
$$(x+a)^2=(x+a)(x+a)$$
Let's write one of the factors $(x+a)=c$, and distribute:
$$(x+a)^2=(x+a)c=xc+ac$$
Replacing $c$ back to $x+a$ we get
$$(x+a)^2=(x+a)(x+a)=x(x+a)+a(x+a)$$
And we distribute again
$$(x+a)^2=x(x+a)+a(x+a)=x^2+xa+ax+a^2$$
At the end we get 
$$(x+a)^2=x^2+2ax+a^2$$
