Let f(n) = 2n^2+5n-3
By the rational roots theorem, if f(n) = 0 has rational roots then they are all of the form p/q in lowest terms, where p is a divisor of 3 and q is a divisor of 2.
Moreover, since 2 only factors as 1 xx 2 (or -1 xx -2), one of the two corresponding linear factors must have q = +-1, so p/q is an integer.
As a result, one of +-1 or +-3 must be a root of f(n) = 0...
f(1) = 2+5-3 = 4
f(-1) = 2-5-3 = -6
f(3) = 18+15-3 = 30
f(-3) = 18-15-3 = 0
So n = -3 is a root and (n+3) is a factor.
The other factor must be (2n-1) in order that the coefficient of n^2 is 2 and the constant term is -3 when these two factors are multiplied.
It's actually quicker to do than to write these words, but we find:
2n^2+5n-3 = (n+3)(2n-1)
