I guess the expressions are supposed to be (<em>a</em> + 1)² and (<em>a</em> + 1)³.
Consider the inequality,
(<em>a</em> + 1)² > (<em>a</em> + 1)³
Move everything to one side:
(<em>a</em> + 1)² - (<em>a</em> + 1)³ > 0
Factorize the left side:
(<em>a</em> + 1)² (1 - (<em>a</em> + 1)) > 0
-<em>a</em> (<em>a</em> + 1)² > 0
<em>a</em> (<em>a</em> + 1)² < 0
Notice that the left side is exactly 0 if either <em>a</em> = 0 or <em>a</em> = -1.
Now consider the following 3 cases:
• If <em>a</em> < -1, then <em>a</em> is negative, while (<em>a</em> + 1)² is always non-negative, which means <em>a</em> (<em>a</em> + 1)² will always be negative.
• If -1 < <em>a</em> < 0, then again <em>a</em> is negative, so <em>a</em> (<em>a</em> + 1)² is also negative.
• If <em>a</em> > 0, then <em>a</em> (<em>a</em> + 1)² is always positive.
So the inequality is satisfied for all <em>a</em> in the interval (-∞, -1) ∪ (-1, 0). This is to say that (<em>a</em> + 1)² is always greater than (<em>a</em> + 1)³ if <em>a</em> is chosen from this domain.
Another way to look at this: assume <em>a</em> + 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 <em>a</em> + 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.
