Range of Real Value Expression from Non Real Complex Number

To determine which value $a$ cannot assume, let $z = x + iy$, where $x, y \in \mathbb{R}$ and $y \neq 0$ because $z$ has a non-zero imaginary part.

Given the equation: $a = z^2 + z + 1$ Rearranging the equation gives a quadratic in $z$: $z^2 + z + (1 - a) = 0$

Using the quadratic formula to solve for $z$: $z = \frac{-1 \pm \sqrt{1 - 4(1)(1 - a)}}{2} = \frac{-1 \pm \sqrt{4a - 3}}{2}$

Since $z$ is a complex number with a non-zero imaginary part ($\text{Im}(z) \neq 0$), the discriminant of this quadratic equation must be strictly negative: $\Delta = 4a - 3 < 0$

Solving this inequality for $a$: $4a < 3 \implies a < \frac{3}{4}$

Thus, for $z$ to have a non-zero imaginary part, $a$ must be strictly less than $\frac{3}{4}$.

Therefore, $a$ cannot assume the value $\frac{3}{4}$.

Hence, the correct option is (D).