First, let's simplify the equation by expanding the terms:
4(x+5)(x+6)(x+10)(x+12) - 3x^2 = 0
Expanding the terms, we get:
4(x^2 + 6x + 5x + 30)(x^2 + 12x + 10x + 60) - 3x^2 = 0
4(x^2 + 11x + 30)(x^2 + 22x + 60) - 3x^2 = 0

Now multiply the terms within the brackets:
4(x^4 + 22x^3 + 60x^2 + 11x^3 + 242x^2 + 660x + 30x^2 + 660x + 1800) - 3x^2 = 0
4(x^4 + 33x^3 + 332x^2 + 1320x + 1800) - 3x^2 = 0

Expanding this further and simplifying, we get:
4x^4 + 132x^3 + 1328x^2 + 5280x + 7200 - 3x^2 = 0

Subtract 3x^2 from both sides:
4x^4 + 132x^3 + 1325x^2 + 5280x + 7200 = 0

Therefore, the solution to the equation 4(x+5)(x+6)(x+10)(x+12) - 3x^2 = 0 is:
4x^4 + 132x^3 + 1325x^2 + 5280x + 7200 = 0