Let $$|\vec{a}| = 2$$, $$|\vec{b}| = 3$$ and the angle between the vectors $$\vec{a}$$ and $$\vec{b}$$ be $$\frac{\pi}{4}$$. Then $$|(\vec{a} + 2\vec{b}) \times (2\vec{a} - 3\vec{b})|^2$$ is equal to

$$(\vec{a} + 2\vec{b}) \times (2\vec{a} - 3\vec{b})$$

$$= 2(\vec{a} \times \vec{a}) - 3(\vec{a} \times \vec{b}) + 4(\vec{b} \times \vec{a}) - 6(\vec{b} \times \vec{b})$$

$$= 0 - 3(\vec{a} \times \vec{b}) - 4(\vec{a} \times \vec{b}) - 0$$

$$= -7(\vec{a} \times \vec{b})$$

$$|(\vec{a} + 2\vec{b}) \times (2\vec{a} - 3\vec{b})|^2 = 49|\vec{a} \times \vec{b}|^2$$

$$|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta = 2 \times 3 \times \sin\frac{\pi}{4} = 6 \times \frac{\sqrt{2}}{2} = 3\sqrt{2}$$

$$|\vec{a} \times \vec{b}|^2 = 18$$

$$49 \times 18 = 882$$

This matches option 4: 882.