Even or odd which one

If $a-b$ is even, then either both $a$ and $b$ are odd or both are even. In both cases, $a+b$ is even. Then $\underbrace{(a+b)^2}_{even} + \underbrace{a+b+1}_{odd} - \underbrace{4}_{even}$ is odd .

Relevant wiki: Parity of Integers

If $a - b$ is even, $a+b$ must be also even. We have,

$(a+b)^2 + (a+b+1) - 4 = (a+b)^2 + a+b - 3$

• $(a+b)(a+b)$ is even since $\text{Even} \times \text{Even} = \text{Even}$

• $a+b-3$ is odd since $\text{Even} \pm \text{Odd} = \text{Odd}$

Hence $(a+b)^2 + a+b - 3$ is odd since $\text{Even} \pm \text{Odd} = \text{Odd}$