Final answer:
The expression (a+1)² is greater than (a+1)³ when a negative value is used for a, including -1, because a negative number plus one is closer to zero and a number between 0 and 1 raised to a higher power becomes smaller.
Step-by-step explanation:
Nita finds that the expression (a+1)² is sometimes not always greater in value than (a+1)³ when evaluating both expressions for the same value of a. To explain her finding, we have to analyze different scenarios for a.
When a is any negative value other than -1, the expression (a+1)² will be greater than (a+1)³. This is because when a is negative, a+1 is closer to zero and raising a number between 0 and 1 to a higher power makes it smaller. For example, if a is -0.5, then (-0.5+1)² = 0.25 and (-0.5+1)³ = 0.125, so the square is greater than the cube.
However, when a is -1, both expressions equal zero. And for a being positive, (a+1)³ will typically be larger because the cubed term increases faster than the squared term as a increases.
Therefore, (a+1)² is greater than (a+1)³ when Nita uses a negative value for a including -1, making the correct option (d) Negative value for a including -1.