Solve the equation 4 2x = -2 for 0^ <= x <= 360^.

Divide both sides of the equation by 4:

Given the range for $x$ is $0^\circ \leq x \leq 360^\circ$.
Multiply the inequality by 2 to find the range for $2x$:

The cosine function is negative in the second and third quadrants.
The reference angle $\alpha$ for which $\cos \alpha = \frac{1}{2}$ is $60^\circ$.
In the second quadrant, $2x = 180^\circ - 60^\circ = 120^\circ$.
In the third quadrant, $2x = 180^\circ + 60^\circ = 240^\circ$.

The general solutions for $\cos \theta = -\frac{1}{2}$ are $\theta = 120^\circ + n \cdot 360^\circ$ and $\theta = 240^\circ + n \cdot 360^\circ$, where $n$ is an integer.

For $2x = 120^\circ + n \cdot 360^\circ$:

• If $n=0$, $2x = 120^\circ$.
• If $n=1$, $2x = 120^\circ + 360^\circ = 480^\circ$.
• If $n=2$, $2x = 120^\circ + 720^\circ = 840^\circ$ (This is outside the range $0^\circ \leq 2x \leq 720^\circ$).

For $2x = 240^\circ + n \cdot 360^\circ$:

• If $n=0$, $2x = 240^\circ$.
• If $n=1$, $2x = 240^\circ + 360^\circ = 600^\circ$.
• If $n=2$, $2x = 240^\circ + 720^\circ = 960^\circ$ (This is outside the range $0^\circ \leq 2x \leq 720^\circ$).

So, the values for $2x$ are $120^\circ, 240^\circ, 480^\circ, 600^\circ$.

Divide each value of $2x$ by 2:

• $x = \frac{120^\circ}{2} = 60^\circ$
• $x = \frac{240^\circ}{2} = 120^\circ$
• $x = \frac{480^\circ}{2} = 240^\circ$
• $x = \frac{600^\circ}{2} = 300^\circ$

The solutions for $x$ in the given range are $60^\circ, 120^\circ, 240^\circ, 300^\circ$.