We are looking to factor $23x^2 + kx - 5.$ Some values of k allow us to factor it as a product of linear binomials with integer coefficients. What are all such values of k?
In order for 23x2+kx−5 to factor as a product of linear binomials with integer coefficients, the two linear terms must have a common factor of 23. This means that k must be a multiple of 23. The only multiples of 23 that work are k=−23 and k=0, so the only possible values of k are −23,0.
Here is the proof of the claim that the two linear terms must have a common factor of 23. Suppose that 23x2+kx−5 can be factored as a product of linear binomials with integer coefficients:
23x2+kx−5=(ax+b)(cx+d)
for some integers a, b, c, and d. Then both ax+b and cx+d must be divisible by 23, since the only common factor of 23x2 and −5 is 1. But this means that k=ad+bc must also be divisible by 23.
