To simplify the expression \( 36x - 216x^2 - 36 \), we can start by factoring out common terms.
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First, observe that all terms can be divided by 36: \[ 36x - 216x^2 - 36 = 36(x - 6x^2 - 1) \]
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Now, we rewrite the expression inside the parentheses: \[ x - 6x^2 - 1 = -6x^2 + x - 1 \]
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Now, factoring \(-6x^2 + x - 1\) can be tricky. To factor it, we can look for two numbers that multiply to \(-6 \times -1 (-6)\) and add up to \(1\). These numbers are \(3\) and \(-2\).
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Rewrite the middle term using these factors: \[ -6x^2 + 3x - 2x - 1 \] Now, factor by grouping: \[ -3x(2x - 1) + 1(2x - 1) \]
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Factor out the common factor \((2x - 1)\): \[ = (2x - 1)(-3x + 1) \]
Putting it all together, we get the complete factorization: \[ 36(x)(2x - 1)(-3x + 1) -36 \]
However, since we need the simplest form of the original expression and we were looking for a simplified version, we notice that:
Going back to \( 36x - 216x^2 - 36 = 36(x - 6x^2 - 1) \)
Now, looking through the options provided, it seems like there may have been a misconception with the simplification process.
So in reviewing, \( 36x-216x^2-36 \) simplified would ultimately not lead to any of the provided response formats which are fractions.
Thus, the simplest form of \( \frac{36}{x} - 6\) is closely achieved.
Since \(36(x - 6x^2 - 1)\) with more assumptions doesn't yield the others listed choices, thus \(36x−6\) is a viable solution noted within responses fitting.
Final response:
36x - 6 would be the simplest interpretation based on the changes.