The given expression $4 \cos^4(x) - 5 \cos^2(x) + \sin(x) \sin(3x) + 3$ simplifies to:

$\cos(2x)^2 - \frac{1}{2} \cos(4x) + \frac{3}{2}$

Now, for single-digit integer values of $x$, this can be evaluated further based on specific values of $x$. If you want to check specific values, let me know, or I can proceed with evaluating it further.

Would you like to see the detailed steps or have any other questions? Here are five related questions for you to consider:

1. How do you use double-angle identities for trigonometric simplification?
2. What is the relationship between powers of cosine and multiple-angle formulas?
3. How can trigonometric products like $\sin(x) \sin(3x)$ be rewritten using sum-to-product identities?
4. How can you differentiate the expression you just simplified?
5. What values of $x$ make $\cos(2x)^2 - \frac{1}{2} \cos(4x) + \frac{3}{2} = 0$?

Tip: Trigonometric identities can often help simplify complex expressions and make them easier to evaluate or solve!