2 (sin³(x) + cos³(x)) - (sin(x) - cos(x))

To simplify the expression \( 2 (\sin^3(x) + \cos^3(x)) - (\sin(x) - \cos(x)) \), we can start by using the identity for the sum of cubes:

\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]

where \( a = \sin(x) \) and \( b = \cos(x) \).

1. **Apply the sum of cubes identity**:
\[
\sin^3(x) + \cos^3(x) = (\sin(x) + \cos(x))(\sin^2(x) - \sin(x)\cos(x) + \cos^2(x))
\]
Since \( \sin^2(x) + \cos^2(x) = 1 \),
\[
\sin^3(x) + \cos^3(x) = (\sin(x) + \cos(x))(1 - \sin(x)\cos(x))
\]

2. **Substituting back**:
Now substituting this into the expression:
\[
2 \left[(\sin(x) + \cos(x))(1 - \sin(x)\cos(x))\right] - (\sin(x) - \cos(x))
\]
Distributing the \( 2 \):
\[
2(\sin(x) + \cos(x))(1 - \sin(x)\cos(x)) - \sin(x) + \cos(x)
\]

3. **Distributing further**:
Expanding:
\[
2(\sin(x) + \cos(x)) - 2(\sin(x) + \cos(x))\sin(x)\cos(x) - \sin(x) + \cos(x)
\]
Combine the terms:
\[
(2\sin(x) + 2\cos(x) - \sin(x) + \cos(x)) - 2(\sin(x) + \cos(x))\sin(x)\cos(x)
\]
Which simplifies to:
\[
(2\sin(x) - \sin(x) + 2\cos(x) + \cos(x)) - 2(\sin(x) + \cos(x))\sin(x)\cos(x)
\]
So it becomes:
\[
(2\sin(x) + 3\cos(x)) - 2(\sin(x) + \cos(x))\sin(x)\cos(x)
\]

At this point, we have simplified the expression, but further simplification might diverge. You can either keep it in this expanded form or evaluate it based on specific values of \( x \).

The final simplified form is:
\[
(2\sin(x) + 3\cos(x)) - 2(\sin(x) + \cos(x))\sin(x)\cos(x)
\]