Solve for x \( 25^{2 x+1}=125^{3 x+4} \) \( x= \)

To solve the equation \( 25^{2x+1} = 125^{3x+4} \), let's first express both sides in terms of powers of 5. We know that \( 25 = 5^2 \) and \( 125 = 5^3 \). Substituting these values, we get: \[ (5^2)^{2x+1} = (5^3)^{3x+4} \] This simplifies to: \[ 5^{4x + 2} = 5^{9x + 12} \] Since the bases are the same, we can set the exponents equal to each other: \[ 4x + 2 = 9x + 12 \] Now, let's solve for \( x \). First, we'll isolate \( x \) on one side: \[ 2 - 12 = 9x - 4x \] \[ -10 = 5x \] \[ x = -2 \] Thus, the solution is: \[ x = -2 \]