How to show that $(-a)(-b)=ab$ holds in a field?

Question

Let $\mathcal{F}$ be a field. How do I prove that $$\forall a,b \in \mathcal{F}:(-a)(-b)=ab.$$

Answer 1

Hint: Proof that $$(-a)(-b)+(-(ab))=0$$

Answer 2

$$(a+(-a))b=0$$ $$ab+(-a)b=0$$ $$(-a)b=-(ab)$$

Now

$$(-a)(b+(-b))=0$$ $$(-a)b+(-a)(-b)=0$$ $$-(ab)+(-a)(-b)=0$$ $$(-a)(-b)=ab$$