2 (sin³(x)) + (cos³(x)) + sin(2x)(sin(x) + cos(x)) = 2√2

To solve the equation

\[
2 (\sin^3(x)) + (\cos^3(x)) + \sin(2x)(\sin(x) + \cos(x)) = 2\sqrt{2}
\]

we can break it down step by step.

1. **Recognize trigonometric identities:**
- \( \sin(2x) = 2\sin(x)\cos(x) \)

2. **Substituting into the equation:**
The equation can be rewritten with this identity:

\[
2\sin^3(x) + \cos^3(x) + 2\sin(x)\cos(x)(\sin(x) + \cos(x)) = 2\sqrt{2}
\]

3. **Simplifying \( \sin(x) + \cos(x) \):**
- Let \( y = \sin(x) + \cos(x) \).
- Remember that \( \sin^2(x) + \cos^2(x) = 1 \).

4. **Finding relationships:**
Now, we will also apply the identity \( \sin^3(x) + \cos^3(x) = (\sin(x) + \cos(x))(\sin^2(x) - \sin(x)\cos(x) + \cos^2(x)) \).

Therefore, you can transform the equation further to express everything in terms of \( y \).

5. **Setting bounds or testing specific values:**
You can substitute specific values for \( x \) (like \( x = \frac{\pi}{4} \), where \( \sin(x) = \cos(x) = \frac{1}{\sqrt{2}} \)).

If \( x = \frac{\pi}{4} \):

\[
2\left(\frac{1}{\sqrt{2}}\right)^3 + \left(\frac{1}{\sqrt{2}}\right)^3 + \sin\left(\frac{\pi}{2}\right)\left(\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}\right)
\]
yields a comparison to \( 2\sqrt{2} \).

6. **Final Solve and Verify:**
Continue with potential simplifications until you reach a point that satisfies the original equation or logically deduce suitable angles.

This method allows you to explore solutions effectively and verifies if specific angles hold true for the equality.