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

(x+4)(4x+6) = x(4x+6) + 4(4x+6) = 4x^2 + 6x + 16x + 24 = 4x^2 + 22x + 24

Next, we can simplify the entire left side of the equation:

(4x^2 + 22x + 24) - (2x+1)^2 = -4

Next, let's expand and simplify the square term:

(2x+1)^2 = (2x+1)(2x+1) = 4x^2 + 2x + 2x + 1 = 4x^2 + 4x + 1

Substitute this back into the equation:

4x^2 + 22x + 24 - (4x^2 + 4x + 1) = -4

This simplifies to:

18x + 23 = -4

Now, we can solve for x:

18x = -27

x = -27/18

x = -3/2

Therefore, the solution to the equation is x = -3/2.