(a)
If a is an odd integer show a² + 3a + 5 is odd
An odd integer is of the form 2k+1 for some integer k.
a = 2k + 1
a² = (2k + 1)² = 4k² + 4k + 1
3a = 6k+3
a² + 3a + 5 = 4k² + 4k + 1 + 6k+3 + 1 = 4k² + 10k + 4 + 1 = 2(2k² + 5k + 2) + 1
Since k is an integer, so is l=2k² + 5k + 2, so a² + 3a + 5 is in the form 2l+1 so is odd.
QED
(b)
a | b
is read a divides b, aka b is a multiple of a. So there's some integer k such that
b = ak
Squaring,
b² = a²k²
That shows b² is an integer multiple of a² so we conclude
a² | b²
QED
