\( \cos^4 \alpha - \sin^4 \alpha - \cos 4\alpha = \sin 3\alpha . \)

To solve the equation \( \cos^4 \alpha - \sin^4 \alpha - \cos 4\alpha = \sin 3\alpha \), we can start by rewriting the terms.

1. **Using trigonometric identities**:
\[
\cos^4 \alpha - \sin^4 \alpha = (\cos^2 \alpha + \sin^2 \alpha)(\cos^2 \alpha - \sin^2 \alpha) = 1 \cdot \cos 2\alpha = \cos 2\alpha.
\]

2. **Recall the identity for \(\cos 4\alpha\)**:
\[
\cos 4\alpha = 2\cos^2 2\alpha - 1.
\]
We can also express \(\sin 3\alpha\):
\[
\sin 3\alpha = 3\sin\alpha - 4\sin^3 \alpha.
\]

3. **Rewrite the original equation**:
The equation becomes:
\[
\cos 2\alpha - \cos 4\alpha = \sin 3\alpha.
\]

4. **Substituting \(\cos 4\alpha\)** gives:
\[
\cos 2\alpha - (2\cos^2 2\alpha - 1) = \sin 3\alpha.
\]
Simplifying this leads to:
\[
1 - \cos 2\alpha - 2\cos^2 2\alpha = \sin 3\alpha.
\]

5. **Final form**:
This is now a trigonometric equation in \(\alpha\). To solve it, we'd typically analyze the left side depending on specific values of \(\alpha\).

Taking inference from trigonometric behaviors:
- Special angles like \(\alpha = 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi\) can be substituted.
- To verify its correctness, evaluate both sides for those values.

This structure allows you to discern possible solutions or behavioral patterns in specific ranges related to \(\alpha\).