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

To solve the inequality, we will follow these steps:

Step 1: Find the critical points of the inequality by setting the numerator and denominator equal to zero.

Numerator: 4x + 1 = 0 4x = -1 x = -1/4

Denominator: 4x + 12 = 0 4x = -12 x = -12/4 x = -3

Step 2: Create a number line and mark the critical points on it.

      -∞    -3        -1/4    +∞

Step 3: Choose test points in each interval to determine the sign of the expression within those intervals.

For the interval (-∞, -3): Let's choose x = -4. When x = -4: (4x + 1)(3 - 6x)/(4x + 12) = (4*(-4) + 1)(3 - 6*(-4))/(4*(-4) + 12) = (-15)(27)/(-4) = 405/4

Since 405/4 is positive, the expression in this interval is positive.

For the interval (-3, -1/4): Let's choose x = 0. When x = 0: (4x + 1)(3 - 6x)/(4x + 12) = (40 + 1)(3 - 60)/(4*0 + 12) = (1)(3)/12 = 1/4

Since 1/4 is positive, the expression in this interval is positive.

For the interval (-1/4, +∞): Let's choose x = 1. When x = 1: (4x + 1)(3 - 6x)/(4x + 12) = (41 + 1)(3 - 61)/(4*1 + 12) = (5)(-3)/16 = -15/16

Since -15/16 is negative, the expression in this interval is negative.

Step 4: Determine the intervals where the expression is less than zero (i.e., negative).

From the test points, we can see that the expression is negative in the interval (-1/4, +∞).

Step 5: Write the solution.

The solution to the inequality is (-1/4, +∞).