Does the sum of two negative integers always negative

The answer is yes. The sum of two negative numbers is always negative.

A negative number can be thought of as debt when you think of numbers in the context of money. For example, the value - 5 or negative 5, can represent the idea that you're 5 dollars in debt. If you go get another 3 dollars in debt, then you'll be 5+3 = 8 dollars in debt total. The "8 dollars in debt" is represented by - 8

So this is what it looks like with negative numbers added up

(-5) + (-3) = - 8

which can also look like this

(-5) + (-3) = - (5+3) = - (8) = - 8