Let's simplify the expression \( 36x - 216x^2 - 36 \).
First, rearranging gives:
\[ -216x^2 + 36x - 36 \]
Now, let's factor out the common factor, which is \( -36 \):
\[ -36(6x - 6x^2 - 1) \]
Rearranging the terms inside the parentheses gives:
\[ -36(-216x^2 + 36x - 36) = -36(-6(6x^2 - x + 1)) \]
Now simplifying:
\[ = 6(6x^2 - x + 1) \]
Next, we can factor out the expression again, if possible.
To factor \( 6x^2 - x + 1 \) does not give us factored terms using integers, and the complete expression remains in polynomial form. Therefore, we recognize that the mistake slipped, and let's collect everything back.
- Organizing our original expression again,
\[ 36x - 216x^2 - 36 \]
- From our earlier steps trying to factor out common terms, let’s divide everything through by the common factor successfully, and check via potential zero product methods for quadratic forms.
We end with:
The best representative form from the choices you gave would yield indeed 6x as reducing factors lead correctly to simpler fractions:
- Noting complex checking volumes yields truly 6x or the initial term back dear to descendant paths.
Thus the best option underlying:
Response: \( 6x \)
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