We are tasked with simplifying the expression:

$4\cos^4x - 5\cos^2x + \sin x \sin(3x) + 3$

Step 1: Simplify $\sin x \sin(3x)$

Using the product-to-sum identity for sine functions, we know that: $\sin A \sin B = \frac{1}{2}[\cos(A - B) - \cos(A + B)]$ For $A = x$ and $B = 3x$, we have: $\sin x \sin(3x) = \frac{1}{2}[\cos(2x) - \cos(4x)]$

So the expression becomes: $4\cos^4x - 5\cos^2x + \frac{1}{2}[\cos(2x) - \cos(4x)] + 3$

Step 2: Rewrite the expression with terms combined

Now substitute the simplified sine product term into the original equation: $4\cos^4x - 5\cos^2x + \frac{1}{2}[\cos(2x) - \cos(4x)] + 3$

Step 3: Analyzing for further simplifications

At this point, the expression can be simplified further depending on specific values for $x$, but there aren't any general identities to simplify all the terms globally without specific information about $x$. The structure of the equation suggests it's the final simplified form in terms of trigonometric identities.

The final expression: $4\cos^4x - 5\cos^2x + \frac{1}{2}[\cos(2x) - \cos(4x)] + 3$

Would you like me to solve this for a particular value of $x$ or further explore specific cases?

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Related questions:

1. How can trigonometric identities be used to simplify complex expressions?
2. What is the product-to-sum identity for sine and cosine?
3. What are some practical applications of trigonometric identities?
4. How do you solve a quartic equation in terms of trigonometric functions?
5. How can we use substitution to simplify $\cos^4 x$ or $\cos^2 x$?

Tip: Memorizing key trigonometric identities like the product-to-sum formulas can make solving such problems faster and more intuitive.