(4x+1)(3-6x)/4x+12<0

To solve this inequality, we need to consider the signs of the factors and determine the regions where the inequality holds true.

First, let's find the critical points, where the factors become zero:

For (4x+1), setting it to zero: 4x + 1 = 0 Solving for x: 4x = -1, x = -1/4

For (4x+12), setting it to zero: 4x + 12 = 0 Solving for x: 4x = -12, x = -12/4, x = -3

So, the critical points are x = -1/4 and x = -3.

Now, let's consider the intervals:

1. Interval (-∞, -3): For this interval, both (4x+1) and (4x+12) are negative, so their product is positive. Therefore, the inequality is not satisfied in this interval.

2. Interval (-3, -1/4): For this interval, (4x+1) is positive, and (4x+12) is negative. The product of a positive and negative number is negative. Therefore, the inequality is satisfied in this interval.

3. Interval (-1/4, +∞): For this interval, both (4x+1) and (4x+12) are positive, so their product is positive. Therefore, the inequality is not satisfied in this interval.

In summary, the solution to the inequality (4x+1)(3-6x)/(4x+12) < 0 is x ∈ (-3, -1/4).