Look for the Even One.

If (a - b) is even, then both a,b are even or both a,b are odd. In either case ab + a is even. The rest are obviously odd.

$(a-b)$ is even mean or $a$ and $b$ also even, or odd. $b$ is possible be odd. $ab$ is possible be odd, when $a$ and $b$ also odd. (tips: $odd \times odd$ is odd.) $a+2b$ is possible be odd, when $a$ is an odd. (tips: $2b$ always even.) $ab +a$ is impossible be odd, because $ab$ and $a$ or even together, or odd together, and that let sum is even. $ab-2a$ is possibel be odd, when $a$ and $b$ also odd. (tips: $2a$ always even.)

given condition is (a-b) is even
eg 1; a(even) - b(even) = even
so,,,,,,,,,ab(even) + a(even) = even
eg 2; a(odd) - b(odd) = even
so,,,,,,,,,ab(odd) + a(odd) = even