Let $z=x+iy,\,\bar z=x-iy$ and we are given $\operatorname{Re}(z)=3\implies x=3.$ So write $z=3+iy,\qquad \bar z=3-iy,\qquad y\in\mathbb R.$

We need the set $\left\{\operatorname{Re}\left(\frac{z-\bar z+z\bar z}{2-3z+5\bar z}\right):\operatorname{Re}(z)=3\right\}.$

1. Simplify numerator and denominator

Numerator

$z-\bar z=(3+iy)-(3-iy)=2iy$ and $z\bar z=(3+iy)(3-iy)=9+y^2.$ Hence $z-\bar z+z\bar z=9+y^2+2iy.$

Denominator

$2-3z+5\bar z=2-3(3+iy)+5(3-iy).$ Now, $2-9-3iy+15-5iy=8-8iy=8(1-iy).$

Thus the expression becomes $\frac{9+y^2+2iy}{8(1-iy)}.$

We need its real part.

2. Compute the real part

Let $w=\frac{9+y^2+2iy}{8(1-iy)}=\frac{1}{8}\cdot \frac{9+y^2+2iy}{1-iy}.$ Multiply numerator and denominator by the conjugate $1+iy$: $w=\frac{1}{8}\cdot \frac{(9+y^2+2iy)(1+iy)}{1+y^2}.$

Now expand the numerator: [ (9+y^2+2iy)(1+iy) =(9+y^2)+i y(9+y^2)+2iy+2i^2y^2. ] Since $i^2=-1$, [ =(9+y^2)-2y^2+i\big(y(9+y^2)+2y\big) =(9-y^2)+i(y^3+11y). ] Therefore, $w=\frac{1}{8}\cdot \frac{(9-y^2)+i(y^3+11y)}{1+y^2}.$ So $\operatorname{Re}(w)=\frac{1}{8}\cdot \frac{9-y^2}{1+y^2}.$

Hence the required set is the range of $f(y)=\frac{9-y^2}{8(1+y^2)},\qquad y\in\mathbb R.$

3. Find the range of $f(y)$

Put $t=y^2\ge 0.$ Then $f(y)=\frac{9-t}{8(1+t)}=:g(t),\qquad t\ge 0.$

Now check monotonicity: $g'(t)=\frac{-(1+t)-(9-t)}{8(1+t)^2}=\frac{-10}{8(1+t)^2}<0.$ So $g(t)$ is strictly decreasing for $t\ge 0$.

Therefore:

• maximum occurs at $t=0$,
• minimum is approached as $t\to\infty$.

Maximum

$g(0)=\frac{9}{8}.$ So the right endpoint is included.

Lower limit

$\lim_{t\to\infty} \frac{9-t}{8(1+t)}=-\frac{1}{8}.$ This value is not attained for any finite $t$, so the left endpoint is open.

Thus the set is $\left(-\frac18,\frac98\right].$ Hence $\alpha=-\frac18,\qquad \beta=\frac98.$

4. Compute $24(\beta-\alpha)$

$\beta-\alpha=\frac98-\left(-\frac18\right)=\frac{10}{8}=\frac54.$ Therefore, $24(\beta-\alpha)=24\cdot \frac54=30.$

5. Match with options

The correct option is: $\boxed{\text{D: }30}$

6. Comparison with stored answer

Stored correct answer: D

Our derived answer is also D. So they agree.