To solve this equation, we need to expand both sides and then simplify to see if they are equal.

Expanding the left side:
(3x-7)^2(x+5) = (3x-7)(3x-7)(x+5)
= (9x^2 - 42x + 49)(x+5)
= 9x^3 - 42x^2 + 49x + 45x^2 - 210x + 245
= 9x^3 + 3x^2 - 161x + 245

Expanding the right side:
(3x-7)(x+5)^2 = (3x-7)(x^2 + 10x + 25)
= 3x^3 + 30x^2 + 75x - 7x^2 - 70x - 175
= 3x^3 + 23x^2 + 5x - 175

Now we will see if both sides are equal:
9x^3 + 3x^2 - 161x + 245 = 3x^3 + 23x^2 + 5x - 175

Simplifying:
9x^3 + 3x^2 - 161x + 245 = 3x^3 + 23x^2 + 5x - 175
6x^3 - 20x^2 + 166x - 420 = 0

Therefore, the original equation is not true because the two sides are not equal.