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Which expression can be used to find the difference of the polynomials?

Check the final answer first, then review the worked steps.

Answer

\((4m - 5) + (-6m + 7 - 2n)\)

Step-by-step solution

  1. Understand the problem: The problem asks to find an equivalent expression for the difference between two polynomials: $(4m - 5)$ and $(6m - 7 + 2n)$. Subtracting a polynomial is the same as adding its additive inverse.
  2. Rewrite the subtraction as addition: To subtract $(6m - 7 + 2n)$ from $(4m - 5)$, we can rewrite the expression as $(4m - 5) - (6m - 7 + 2n)$. This is equivalent to adding the opposite of the second polynomial: $(4m - 5) + (-(6m - 7 + 2n))$.
  3. Distribute the negative sign: Distribute the negative sign to each term inside the second parenthesis: $-(6m - 7 + 2n) = -6m + 7 - 2n$.
  4. Form the equivalent addition expression: Now, substitute this back into the expression from step 2: $(4m - 5) + (-6m + 7 - 2n)$. This expression represents the difference of the original polynomials.
  5. Combine like terms (optional, for verification): To verify, we can simplify this expression. Combine the $m$ terms: $4m - 6m = -2m$. Combine the constant terms: $-5 + 7 = 2$. Combine the $n$ terms: $-2n$. So, the simplified difference is $-2m + 2 - 2n$. The expression $(4m - 5) + (-6m + 7 - 2n)$ is the correct representation of the difference.