Let's simplify the equation step by step:

5(5^(2x)) + 43(5^x) + 24 = 0

Let's substitute y = 5^x. Then we have:

5(y^2) + 43(y) + 24 = 0

Now we have a quadratic equation in terms of y:

5y^2 + 43y + 24 = 0

Let's factor this quadratic equation:

(5y + 8)(y + 3) = 0

Therefore, the solutions for y are y = -8/5 and y = -3.

Now we can substitute back y = 5^x:

5^x = -8/5 and 5^x = -3

These equations do not have real solutions because 5 raised to any power will always be positive. Therefore, the original equation 5(5^(2x)) + 43(5^x) + 24 = 0 has no real solutions.