First, let's simplify the left side of the inequality:

4(x+3) - (5x+1)(2x+6) < 6
4x + 12 - (10x^2 + 30x + 5x + 6) < 6
4x + 12 - 10x^2 - 35x - 6 < 6
4x + 12 - 10x^2 - 35x - 6 < 6
-10x^2 - 31x + 6 < 6

Now, let's set the inequality equal to 0:

-10x^2 - 31x + 6 - 6 < 0
-10x^2 - 31x < 0
x(10x + 31) > 0

Now we need to find the critical points by setting each factor equal to zero:

x = 0 and 10x + 31 = 0
10x = -31
x = -31/10

So, the critical points are x = 0 and x = -31/10. Now we need to test the intervals created by these points by plugging in a test value:

For x < -31/10, let's use x = -4:
(-4)(10(-4) + 31) = (-4)(-9) = 36 > 0

For -31/10 < x < 0, let's use x = -1:
(-1)(10(-1) + 31) = (-1)(21) = -21 < 0

For x > 0, let's use x = 1:
(1)(10(1) + 31) = (1)(41) = 41 > 0

Therefore, the solution to the inequality is x < -31/10 or x > 0.