I guess the expressions are supposed to be (a + 1)² and (a + 1)³.
Consider the inequality,
(a + 1)² > (a + 1)³
Move everything to one side:
(a + 1)² - (a + 1)³ > 0
Factorize the left side:
(a + 1)² (1 - (a + 1)) > 0
-a (a + 1)² > 0
a (a + 1)² < 0
Notice that the left side is exactly 0 if either a = 0 or a = -1.
Now consider the following 3 cases:
• If a < -1, then a is negative, while (a + 1)² is always non-negative, which means a (a + 1)² will always be negative.
• If -1 < a < 0, then again a is negative, so a (a + 1)² is also negative.
• If a > 0, then a (a + 1)² is always positive.
So the inequality is satisfied for all a in the interval (-∞, -1) ∪ (-1, 0). This is to say that (a + 1)² is always greater than (a + 1)³ if a is chosen from this domain.
Another way to look at this: assume a + 1 falls between -1 and 1. Whenever you scale a number between -1 and 1 by another number in the same range, the product will always be smaller than the original number.
More concretely, let's say a + 1 = 1/2. So for instance, 1/2 * 1/2 = (1/2)² = 1/4, and 1/4 is clearly smaller than 1/2. If we multipy again by 1/2, we get 1/2 * 1/2 * 1/2 = (1/2)³ = 1/8, which is smaller than 1/4.
