Prove that (2m + 1)^2 − (2n – 1)^2 = 4(m + n)(m − n + 1)

To demonstrate the given equation, we can start by expanding both sides of the equation. First, we expand the left-hand side of the equation, which is (2m + 1)^2 − (2n – 1)^2. Using the formula (a + b)^2 = a^2 + 2ab + b^2, we can expand (2m + 1)^2 to get 4m^2 + 4m + 1. Similarly, we can expand (2n – 1)^2 to get 4n^2 – 4n + 1. Therefore, the left-hand side of the equation becomes 4m^2 + 4m + 1 – (4n^2 – 4n + 1).

Next, we simplify the expression by combining like terms. We can combine the constants 1 and -1, which cancel each other out, leaving us with 4m^2 + 4m – (4n^2 – 4n). This simplifies further to 4m^2 + 4m – 4n^2 + 4n.

Finally, we can factor the right-hand side of the equation, which is 4(m + n)(m − n + 1). By factoring out the common terms, we get 4(m^2 – n^2 + m + n). Comparing the factored form of the right-hand side with the simplified form of the left-hand side, we can see that they are equal. Therefore, we have successfully proven that (2m + 1)^2 − (2n – 1)^2 = 4(m + n)(m − n + 1).

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