To simplify this expression, we will first expand the terms using the formula for the difference of cubes:

(a^3 - b^3) = (a - b)(a^2 + ab + b^2)

In this case, we have (x+5)^3 -(x+1)^3:

= [(x + 5) - (x + 1)][(x + 5)^2 + (x + 5)(x + 1) + (x + 1)^2]
= (4)(x^2 + 10x + 25 + x^2 + 6x + 5 + x^2 + 2x + 1)
= 4(3x^2 + 18x + 31)

Now we want to simplify 4*(3x^2 - 5):

= 43x^2 - 45
= 12x^2 - 20

Therefore, equating the two expressions from above, we have:

4(3x^2 + 18x + 31) = 12x^2 - 20

Solving for x, we get:

12x^2 + 72x + 124 = 12x^2 - 20
72x + 124 = -20
72x = -144
x = -2

So the solution is x = -2.