To simplify the given expression, let's rewrite it in a more readable format:

$\frac{(x^2 + 2)^3 (26x) - x^2 \cdot 3 (x^2 + 2)^2 (26x)}{[(x^2 + 2)^3]^2}$

Step-by-step Solution:

1. Simplify the Denominator:

   The denominator is:

   $[(x^2 + 2)^3]^2 = (x^2 + 2)^6$

2. Factor Out Common Terms in the Numerator:

   The numerator is:

   $(x^2 + 2)^3 (26x) - x^2 \cdot 3 (x^2 + 2)^2 (26x)$

   We can factor out $(x^2 + 2)^2 \cdot 26x$:

   $= (x^2 + 2)^2 \cdot 26x \left[(x^2 + 2) - 3x^2\right]$

3. Simplify the Expression Inside the Brackets:

   $(x^2 + 2) - 3x^2 = -2x^2 + 2$

   We can factor out -2:

   $= -2(x^2 - 1)$

   Which further factors into:

   $= -2(x-1)(x+1)$

4. Put Everything Together:

   Substituting back, the numerator becomes:

   $= (x^2 + 2)^2 \cdot 26x \cdot -2(x-1)(x+1)$

   $= -52x(x^2 + 2)^2(x-1)(x+1)$

5. Simplify the Whole Expression:

   Now the entire expression can be written as:

   $\frac{-52x(x^2 + 2)^2(x-1)(x+1)}{(x^2 + 2)^6}$

   Simplifying $(x^2 + 2)^2$ from both numerator and denominator:

   $= \frac{-52x(x-1)(x+1)}{(x^2 + 2)^4}$

Final Answer:

$\frac{-52x(x-1)(x+1)}{(x^2 + 2)^4}$

This is the fully simplified and factored form of the given expression.

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

Relative Questions:

1. How do you handle expressions with higher powers when simplifying?
2. Why is it helpful to factor out common terms in the numerator?
3. What are some common mistakes to watch out for in algebraic simplification?
4. How does factoring help in finding the roots of an expression?
5. How would you integrate an expression like the one given here?

Tip:

Always check for common factors in both the numerator and the denominator early in your simplification process, as this can often lead to significant simplifications.