help

If (x + 1)(y + 1) = 40 and xy(x + y) = 380, then find all possible values of x^2 + y^2.

I'm only assuming integer values for  x , y

(x + 1) ( y + 1)  =  40             xy ( x + y)  =  380     (2)

xy  + x + y  +  1  = 40

xy + x + y  =  39

x + y  =  39  - xy    (1)

Sub (1)  into (2)  and we have that

xy ( 39 - xy)  =  380

-(xy)^2 + 39(xy)  = 380

-(xy)^2  + 39(xy) - 380  =  0   multiply through by  -1

(xy)^2  - 39xy  + 380   =  0       factor

(xy  - 20) ( xy - 19)  =  0

Set both factors to  0   and we have that

xy  = 20         xy  = 19

The possible values for  x, y  are

x         y

1       19

19      1

1       20

2       10

4        5

5        4

10      2

20      1

Because  ( x + 1)  ( y  + 1)  =  40

Then  only

(1, 19)   and (19, 1)  will work

So

x^2 + y^2  =   362