Step-by-step Solution:

Here is the MathJax code for the given solution: \[ \textbf{Concept:} \] \[ \text{For the least value of the function, the sum of the square of the roots of the equation } x^2 - (a - 2)x - (a + 1) = 0 \text{ is zero.} \] \[ \textbf{Calculations:} \] \[ \text{Consider, the roots of the equation } x^2 - (a - 2)x - (a + 1) = 0 \text{ are } \alpha \text{ and } \beta \] \[ \Rightarrow \alpha + \beta = a - 2 \quad \text{and} \quad \alpha \beta = -(a + 1) \] \[ \text{Given, the sum of the squares of the roots is the least value:} \] \[ \alpha^2 + \beta^2 = 0 \] \[ \Rightarrow (\alpha + \beta)^2 - 2\alpha\beta = 0 \] \[ \Rightarrow (a - 2)^2 + 2(a + 1) = 0 \] \[ \Rightarrow a^2 - 2a + 4 + 2 = 0 \] \[ \Rightarrow a^2 - 2a + 6 = 0 \] \[ \Rightarrow a^2 - 2a + 1 + 5 = 0 \] \[ \Rightarrow (a - 1)^2 + 5 = 0 \] \[ \Rightarrow a = 1 \]