a and b are integers with a < 0 < b. Which of these expressions is negative for EVERY such pair?

a is negative and $b$ is positive, so $a-b$ is "negative minus positive", which can only be negative: $(-2)-5=-7$ and $(-9)-1=-10$. The others all fail. $a+b$ can go either way, since $(-2)+5=3$ is positive while $(-9)+1=-8$ is negative. $b-a$ is "positive minus negative", so it is always POSITIVE. And $a^2+b$ is a positive square plus a positive number, so it is always positive too.