Let's first expand both sides of the equation to simplify it:

Expanding the left side:
(3x + 15)^2 = (3x + 15)(3x + 15)
= 9x^2 + 45x + 45x + 225
= 9x^2 + 90x + 225

Expanding the right side:
(3x - 9)^2 = (3x - 9)(3x - 9)
= 9x^2 - 27x - 27x + 81
= 9x^2 - 54x + 81

Now we have:
9x^2 + 90x + 225 = 9x^2 - 54x + 81

Subtracting 9x^2 from both sides:
90x + 225 = -54x + 81
144x + 225 = 81
144x = -144
x = -1

So, the solution for the first equation is x = -1.

Now let's solve the second equation:

(x - 5)(x - 3) + 1 = 0
Expanding:
x^2 - 3x - 5x + 15 + 1 = 0
x^2 - 8x + 16 = 0
(x - 4)(x - 4) = 0
(x - 4)^2 = 0
x - 4 = 0
x = 4

Therefore, the solutions for the second equation are x = 4.

In conclusion, the solutions to the system of equations are x = -1 and x = 4.