To factorize the expression:

$3\sin^3 x \cos x + 4\cos^4 x - 3\cos^2 x \sin^2 x - \cos^4 x - \sin x \cos^3 x$

Step 1: Combine like terms

First, let's group similar terms:

$= (3\sin^3 x \cos x - \sin x \cos^3 x) + (4\cos^4 x - \cos^4 x) - 3\cos^2 x \sin^2 x$

Simplify:

$= 3\sin^3 x \cos x - \sin x \cos^3 x + 3\cos^4 x - 3\cos^2 x \sin^2 x$

Step 2: Factor common terms

Now, factor out the common terms where possible:

First group: $3\sin^3 x \cos x - \sin x \cos^3 x$

Factor out $\sin x \cos x$:

$= \sin x \cos x (3\sin^2 x - \cos^2 x)$

Second group: $3\cos^4 x - 3\cos^2 x \sin^2 x$

Factor out $3\cos^2 x$:

$= 3\cos^2 x (\cos^2 x - \sin^2 x)$

Step 3: Combine all the factors

Now the expression is:

$= \sin x \cos x (3\sin^2 x - \cos^2 x) + 3\cos^2 x (\cos^2 x - \sin^2 x)$

Step 4: Factor further

We observe that we have terms with $\cos^2 x - \sin^2 x$, which suggests we might factor using the identity $\cos^2 x - \sin^2 x = \cos(2x)$. So, we rewrite the expression:

$= \sin x \cos x (3\sin^2 x - \cos^2 x) + 3\cos^2 x \cos(2x)$

At this point, there isn't a simple factorization that can combine both terms further in a way that simplifies the expression fully without advanced techniques. Thus, the most factored form that seems most appropriate is:

$= \sin x \cos x (3\sin^2 x - \cos^2 x) + 3\cos^2 x \cos(2x)$

Would you like more details or have any other questions about this problem?

Here are some related questions you can explore:

1. What is the trigonometric identity for $\cos^2 x - \sin^2 x$?
2. How do we expand and factor trigonometric expressions effectively?
3. What is the role of the double angle formula in simplifying trigonometric expressions?
4. Can you solve for $x$ in the equation after factoring?
5. How do you approach factorizing trigonometric polynomials with higher powers?

Tip: Always look for common factors and use standard trigonometric identities like $\sin^2 x + \cos^2 x = 1$ and $\cos(2x) = \cos^2 x - \sin^2 x$ to simplify complex trigonometric expressions.