Answer:
![n=241](https://tex.z-dn.net/?f=n%3D241)
Step-by-step explanation:
We are given
![5x^2+nx+48](https://tex.z-dn.net/?f=5x%5E2%2Bnx%2B48)
Let's assume it can be factored as
![5x^2+nx+48=(5x-s)(x-r)](https://tex.z-dn.net/?f=5x%5E2%2Bnx%2B48%3D%285x-s%29%28x-r%29)
now, we can multiply right side
and then we can compare it
![5x^2+nx+48=5x^2-5rx-sx+rs](https://tex.z-dn.net/?f=5x%5E2%2Bnx%2B48%3D5x%5E2-5rx-sx%2Brs)
![5x^2+nx+48=5x^2-(5r+s)x+rs](https://tex.z-dn.net/?f=5x%5E2%2Bnx%2B48%3D5x%5E2-%285r%2Bs%29x%2Brs)
now, we can compare coefficients
![rs=48](https://tex.z-dn.net/?f=rs%3D48)
![5r+s=-n](https://tex.z-dn.net/?f=5r%2Bs%3D-n)
![n=-(5r+s)](https://tex.z-dn.net/?f=n%3D-%285r%2Bs%29)
now, we can find all possible factors of 48
and then we can assume possible prime factors of 48
![48=-+(1\times 48)](https://tex.z-dn.net/?f=48%3D-%2B%281%5Ctimes%2048%29)
![48=-+(2\times 24)](https://tex.z-dn.net/?f=48%3D-%2B%282%5Ctimes%2024%29)
![48=-+(3\times 16)](https://tex.z-dn.net/?f=48%3D-%2B%283%5Ctimes%2016%29)
![48=-+(4\times 12)](https://tex.z-dn.net/?f=48%3D-%2B%284%5Ctimes%2012%29)
![48=-+(6\times 8)](https://tex.z-dn.net/?f=48%3D-%2B%286%5Ctimes%208%29)
Since, we have to find the largest value of n
So, we will get consider larger value of r because of 5r
and because n is negative of 5r+s
so, we will both n and r as negative
So, we can assume
r=-48 and s=-1
so, we get
![n=-(5\times -48-1)](https://tex.z-dn.net/?f=n%3D-%285%5Ctimes%20-48-1%29)
![n=241](https://tex.z-dn.net/?f=n%3D241)