10 is always even and 6x is always even so they don't interfere in the question so the only relevant part of the equation is x^2:
if x is odd, x^2 is odd
if x is even, x^2 is even

the condition (1) alone is enough to answer the question (if x^2+4x+5 is odd x is even, if x^2+4x+5 is even x is odd)

the condition (2) alone is enought to answer the question. in this case 2*x^2 is always even, as well as 4 so if the equation is even, 3x needs to also be even, which means that x is also even