Final answer:
To ensure the equation ax + a² = 0 has exactly one solution in the interval (-1, 2), the value of 'a' must fall within the open interval (-2, 1), excluding the endpoints.
Step-by-step explanation:
The student is asking for the values of a for which the equation ax + a² = 0 has exactly one solution in the interval (-1, 2). In order to have exactly one solution for any value of x within that interval, the equation must be linear and not quadratic, since a quadratic equation typically has two solutions.
This means that a cannot be zero, as it would not provide a linear equation. Also, the only scenario where a linear equation has a single solution is when it has a unique root that lies within the interval in question. Therefore, the linear equation derived from this would be written as ax = -a² or x = -a.
For this equation to have a single solution within (-1, 2), a must be such that -a falls between -1 and 2. This means that a must be between -2 and 1 exclusively; any values within that open interval would satisfy the condition, but the bounds themselves will not since the interval is open and doesn't include the endpoints.