Find the sum of 3 positive integers whose product is 144, if the sum of their squares is 149.

you can simply solve this with 'trial and error', but I am not sure this is the kind of answer you are looking for...
I would think of factoring 144 as a first step
2*72
3*48
4*36
6*24
8*18
9*16
12*12

(5,7,10,11 are not considered because 144 divided by any of them gives decimal number not integer )

second step would be to factor each of the above options into three values and use trial and error to check which satisfies the second condition (sum of their squares is 149)

e.g:
2*8*9
3*8*6
...
after checking you will find out that 9*16 (9*8*2) satisfies the second condition also (9^2 + 8^2 + 2^2 = 149)

now back to the question ( sum of the integers )

Answer = 9+8+2 = 19