Let's simplify the given expression step by step:
The expression is: \[ -2x(m + n - 4) + 5(-2m + 2n) + n(m + 4n - 5) \]
Since there's no \( x \) present when evaluating the other terms, we can ignore the \(-2x(m + n - 4)\) part for simplification purposes in this context. Let's simplify the other parts first.
Step 1: Simplify each term
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First term: \[ 5(-2m + 2n) = -10m + 10n \]
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Second term: \[ n(m + 4n - 5) = nm + 4n^2 - 5n \]
Step 2: Combine all terms
Now we combine:
- Ignore the \(-2x(m+n-4)\): It's not affecting the other components.
- From the first term, we have \(-10m + 10n\).
- From the second term, we have \(nm + 4n^2 - 5n\).
So, we combine: \[ -10m + 10n + nm + 4n^2 - 5n \]
Step 3: Combine like terms
Now combine the coefficients:
- The \(n\) terms: \(10n - 5n = 5n\)
- The expression simplifies to: \[ -10m + nm + 4n^2 + 5n \]
Step 4: Rearranging
Write it in a conventional form: \[ -10m + nm + 5n + 4n^2 \] which can also be written as: \[ -10m + nm + 4n^2 + 5n \]
Step 5: Identifying equivalent expressions
Now let's examine the given options:
- \(-10m^2 + 4n^2 - mn - 2m + 5n\)
- \(-10m^2 + 4n^2 - 3mn - 2m + 5n\)
- \(-2^2 + 4n^2 - mn - 2m + 5n\)
- \(-2n^2 + 4n^2 - 3mn + 2m + 5n\)
None of the options match our derived expression of: \[ nm + 4n^2 - 10m + 5n \] Let's note that we have \(mn\) rather than \(-mn\), \( -10m\) rather than \( -10m^2\), and so forth.
After re-verifying the individual terms and simplifications, none of these options properly correspond to the derived expression. Double-checking or additional context might be necessary to resolve this. Would you like to explore this further?