Form a quadratic equation with integer coefficients given its roots are -2.5 and 3.

The given roots are $x_1 = -2.5$ and $x_2 = 3$.

A quadratic equation can be written in the form $x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0$.
Substituting the sum and product:

To express the equation with integer coefficients, multiply the entire equation by 2:

The equation is in the form $ax^2 + bx + c = 0$.

The quadratic equation is $\boxed{2x^2 - x - 15 = 0}$.

The equation is $\sin(2x + 10^\circ) = \frac{\sqrt{3}}{2}$.
Let $\theta = 2x + 10^\circ$.
The principal value for $\theta$ is $\arcsin\left(\frac{\sqrt{3}}{2}\right) = 60^\circ$.

Since $\sin \theta$ is positive, $\theta$ lies in the first or second quadrant.
The general solutions for $\sin \theta = \frac{\sqrt{3}}{2}$ are:

where $n$ is an integer.

The given range for $x$ is $0^\circ \leq x \leq 360^\circ$.
We need to find the range for $\theta = 2x + 10^\circ$:
$2(0^\circ) + 10^\circ \leq 2x + 10^\circ \leq 2(360^\circ) + 10^\circ$
$10^\circ \leq \theta \leq 730^\circ$.

Now, list the values of $\theta$ within this range:
For $n=0$:
$\theta = 60^\circ$
$\theta = 120^\circ$

For $n=1$:
$\theta = 60^\circ + 360^\circ = 420^\circ$
$\theta = 120^\circ + 360^\circ = 480^\circ$

For $n=2$:
$\theta = 60^\circ + 2 \cdot 360^\circ = 780^\circ$ (This is outside the range $10^\circ \leq \theta \leq 730^\circ$)
$\theta = 120^\circ + 2 \cdot 360^\circ = 840^\circ$ (This is outside the range $10^\circ \leq \theta \leq 730^\circ$)

So, the valid values for $2x + 10^\circ$ are $60^\circ, 120^\circ, 420^\circ, 480^\circ$.

Case 1: $2x + 10^\circ = 60^\circ$
$2x = 50^\circ$
$x = 25^\circ$

Case 2: $2x + 10^\circ = 120^\circ$
$2x = 110^\circ$
$x = 55^\circ$

Case 3: $2x + 10^\circ = 420^\circ$
$2x = 410^\circ$
$x = 205^\circ$

Case 4: $2x + 10^\circ = 480^\circ$
$2x = 470^\circ$
$x = 235^\circ$

All these values are within the range $0^\circ \leq x \leq 360^\circ$.

The solutions are $\boxed{x = 25^\circ, 55^\circ, 205^\circ, 235^\circ}$.