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Define the function N on Z[√ -5] by N(x) = |x|^2. Note that N(xy)=N(x)N(y), and that

N(a + b√ -5) = a^2 + 5b^2.

In particular, N(x) is always a nonnegative integer. If N(a+b√ -5)=1, then clearly b=0, and a=±1. Therefore N(x)=1 implies x=±1. Similarly, if N(a+b√ -5)=2, then b=0; since 2 is not a square, this shows that N(x) can never be 2.

Suppose 1 + √ -5 = xy for some x,y in Z[√ -5]. Then

6 = N(1 + √ -5) = N(xy) = N(x)N(y).

Since N(x) cannot be 2, it follows that N(x)=1 and N(y)=6 (without loss of generality). Thus x = ±1. In particular, x is a unit, which proves that 1 + √ -5 is irreducible.

You can see that 1 + √ -5 is not prime because

(1 + √ -5)(1 - √ -5) = 6 = 2*3,

but 1 + √ -5 does not divide 2 or 3. You can see that 1 + √ -5 does not divide 2 or 3 because N(1 + √ -5)=6 which does not divide N(2)=4 or N(3)=9.