Algebra

(x + 4) / (x + 5)  =  (x - 5 + 7)  / (2x)

(x + 4) / ( x + 5)  = ( x +2) / (2x)      cross-multiply

(2x) ( x + 4) =  (x + 5) ( x + 2)

2x^2 + 8x =   x^2  + 7x + 10        rearrange as

x^2 + x   =  10        complete the square on x

x^2 + x + 1/4 =  10 + 1/4

(x + 1/2)^2 =  41/4       take both roots

x + 1/2  =  ±  √41 / 2

x =  -1/2 ±  √41 / 2