We are given the expression:

$\frac{4x^3 \cdot (x - 1)(x - 3)^3 - x^4 \cdot (x - 3)^3(4x - 6)}{[(x - 1)(x - 3)^3]^2}$

To simplify this, let's follow step-by-step:

Step 1: Factor out common terms

Both the numerator and the denominator have common factors of $(x - 1)$ and $(x - 3)^3$.

• In the numerator, the common factor is $(x - 1)(x - 3)^3$, so we can factor that out: $(x - 1)(x - 3)^3 \left( 4x^3 - x^4(4x - 6) \right)$

• The denominator is: $[(x - 1)(x - 3)^3]^2 = (x - 1)^2(x - 3)^6$

Step 2: Cancel common terms

We can cancel out $(x - 1)(x - 3)^3$ from both the numerator and the denominator. This leaves us with:

$\frac{4x^3 - x^4(4x - 6)}{(x - 1)(x - 3)^3}$

Step 3: Expand the remaining terms in the numerator

Expand $x^4(4x - 6)$: $x^4(4x - 6) = 4x^5 - 6x^4$

Now, substitute this back into the numerator: $4x^3 - (4x^5 - 6x^4) = 4x^3 - 4x^5 + 6x^4$

So, the numerator becomes: $-4x^5 + 6x^4 + 4x^3$

Step 4: Final expression

Now the expression is simplified to: $\frac{-4x^5 + 6x^4 + 4x^3}{(x - 1)(x - 3)^3}$

This is the simplified version of the given expression.

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Would you like more details on any step? Here are 5 related questions for practice:

1. How can factoring common terms simplify complex expressions?
2. What are the rules for canceling terms in rational expressions?
3. How do you expand products of polynomials?
4. What would happen if you squared both sides of an equation?
5. How do you recognize when terms can be canceled from the numerator and denominator?

Tip: Always look for common factors in both the numerator and denominator to simplify rational expressions quickly.