Question

find the largest value of $n$ such that $5x^2 nx 48$ can be factored as the product of two linear factors with integer coefficients.

Answer 1

If we can write

`5x^2 + nx + 48 = (ax + b) (cx + d)`

then expanding the right side gives

`5x^2 + nx + 48 = acx^2 + (ad+bc)x + bd`

so that
`ac=5`,
`ad+bc=n`, and
`bd=48`, where
`a,b,c,d` are integers. We want to maximize
`ad+bc`.

5 is prime, so
`(a,c)` can be either (1, 5) or (5, 1). Let
`a=5` and
`b=1`. Then maximizing

`ad+bc=5d + c`

is just a matter of picking the largest possible value for
`d`, which is 48. Then
`\max\{ad+bc\}=5*48+1*1=\boxed{241}`.