Prove 5y^2 + 5y + 1 is prime, where y is integer >= 1

This is exercise 8.e of Chapter 1 of "A Concise Introduction to Pure Mathematics"

Prove 5y^2 + 5y + 1 is prime, where y is integer >= 1

suppose if statement is false, then
5y^2 + 5y+ 1 can be written as
(ay + b)(cy + d)
where (ay + b) and (cy + d) are both integers,
hence a, b, c, d has to be rational numbers
extend it we get:
ac*y + (ad + cb) y + bd
so
ac = 5
ad + cb = 5
bd = 1
bd= 1 => b = 1/d, c = 5/a
then ad +cb = 5 => ad + 5/ad = 5
=> (ad)^2 - 5ad + 5 = 0
make ad = e, e is also a rational number
but e = (5 +- sqrt(25 - 4*5))/2 = (5 +- sqrt(5))/2
sqrt(5) is not a rational number, so e is not a rational number
this is a contradiction