My teacher said that, No one knows of any quadratic polynomial that produces an infinite amount of primes. I was thinking could we use a polynomial like
$x^2+1$ and then do a trick similar to Euclids proof of the infinite amount of primes
and assume their are only finitely many of them, But this probably wont work.
How else could we try to do this.

pwsnafu
My teacher said that, No one knows of any quadratic polynomial that produces an infinite amount of primes.
What does this statement mean? How does a quadratic polynomial "produce" a prime?

2^2+1=5 thats what I mean, are values for x are the naturals

Okay, just clarifying, are you asking for a quadratic such that there are an infinite number of positive integer inputs for x which produce prime numbers?

a couple things I noticed is polynomials of the form $x^2-x+1$
will always produce odd numbers and cant be factored so thats a good start.
and the polynomial $x^2+x+1$ produced the same primes as
$x^2-x+1$ Maybe we could find a set of polynomials that covered a large portion of the odd numbers and then we would know at least one of these produced an
infinite amount of primes.