(sin x + cos x)² + (sin² x + (sin²x - cos² x)²

To simplify the expression \((\sin x + \cos x)^2 + \left(\sin^2 x + \left(\sin^2 x - \cos^2 x\right)^2\right)(\sin x + \cos x)^2 + \left(\sin^2 x + \left(\sin^2 x - \cos^2 x\right)^2\right)\), we can follow these steps:

1. **Let’s denote \(A = \sin x + \cos x\)** and \(B = \sin^2 x - \cos^2 x\).
2. Rewrite the expression using \(A\) and \(B\):
\[
A^2 + \left(\sin^2 x + B^2\right)A^2 + \left(\sin^2 x + B^2\right)
\]
3. Now, we need a few identities to simplify \(A\) and \(B\):
- \(A^2 = \sin^2 x + \cos^2 x + 2\sin x \cos x = 1 + 2\sin x \cos x\)
- \(B = \sin^2 x - \cos^2 x\)

4. Substitute \(A^2\) and \(B\):
\[
A^2 + \left(\sin^2 x + B^2\right)A^2 + \left(\sin^2 x + B^2\right)
\]

5. First, we compute \(B^2\):
\[
B^2 = (\sin^2 x - \cos^2 x)^2 = \sin^4 x - 2\sin^2 x \cos^2 x + \cos^4 x
\]

6. Using the identity \(\sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x = 1 - 2\sin^2 x \cos^2 x\):
\[
B^2 = \left(1 - 2\sin^2 x \cos^2 x\right) - 2\sin^2 x \cos^2 x = 1 - 4\sin^2 x \cos^2 x
\]

7. Substitute back into the main expression:
\[
A^2 + \left(\sin^2 x + \left(1 - 4\sin^2 x \cos^2 x\right)\right)A^2 + \left(\sin^2 x + \left(1 - 4\sin^2 x \cos^2 x\right)\right)
\]

8. Simplify the components:
\[
\sin^2 x + 1 - 4\sin^2 x \cos^2 x
\]
This gives us:
\[
(A^2 + \sin^2 x + 1 - 4\sin^2 x \cos^2 x)A^2 + \left(\sin^2 x + 1 - 4\sin^2 x \cos^2 x\right)
\]

9. Now, collect and simplify terms:
- Note that you have \(A^2\) from earlier and the combined terms from \(B\).

10. The challenge is in combining like terms which involve \(2\sin^2 x \cos^2 x\) and those that yield constants.

To finalize and get an exact answer, numerical evaluation at specific angles or constants can help in obtaining clear results, but you will get polynomials involving trigonometric identities such as those surrounding \(\sin^2\) and \(\cos^2\).

Lastly, consider that the simplification doesn’t lead directly to a single trigonometric function or value without specific angle inputs or target constant forms.