To solve this equation, we can first make a substitution to simplify it.
Let y = (5x+6)^2
Therefore, the equation becomes:
y^2 + 5y - 6 = 0

Now, we can factor this quadratic equation:
(y + 6)(y - 1) = 0

This gives us two possible values for y:
y = -6 or y = 1

Now, we substitute back in our original substitution:
(5x+6)^2 = -6 or (5x+6)^2 = 1

For (5x+6)^2 = -6 to be true, it is impossible since the square of a real number cannot be negative. Therefore, we discard it.

For (5x+6)^2 = 1:
Take the square root of both sides:
5x+6 = ±√1
5x+6 = ±1

Now solve for x:
for 5x + 6 = 1:
5x = 1 - 6
5x = -5
x = -1

for 5x + 6 = -1:
5x = -1 - 6
5x = -7
x = -7/5

Therefore, the solutions to the equation are x = -1 and x = -7/5.