Prove equality with non-disjoint subset of $\mathbb R$

Question

Given $\alpha_0$ a generic probability measure, $A,B$ non-disjoint subset of $\mathbb R$ i have to prove the equality: $$\alpha_0(A\cap B)(1-\alpha_0(A \cap B))-\alpha_0(A\cap B)\alpha_0(B \setminus A) -\alpha_0(A\setminus B)\alpha_0(A \cap B) -\alpha_0(A\setminus B) \alpha_0(B\setminus A)= \alpha_0(A\cap B)-\alpha_0(A)\alpha_0(B)$$ I'm really confused about $A\setminus B$ and i dont know how handle it

Answer 1

Hint: You basically need to prove that $$\alpha_0(A \cap B)^2+\alpha_0(A\setminus B)\alpha_0(A \cap B)+\alpha_0(B\setminus A)\alpha_0(A \cap B)+\alpha_0(A\setminus B)\alpha_0(B\setminus A)=\alpha_0(A)\alpha_0(B)$$

by writing $x^2+ax+bx+ab$ in the form $(x+a)(x+b)$.