The given equation is:

$2(25)^{x+1} + 5^{x+2} - 3 = 0$

Step-by-Step Solution:

Step 1: Simplify the powers of 25 and 5

Recall that $25 = 5^2$. So we can rewrite $25^{x+1}$ as follows:

$25^{x+1} = (5^2)^{x+1} = 5^{2(x+1)} = 5^{2x+2}$

Thus, the equation becomes:

$2(5^{2x+2}) + 5^{x+2} - 3 = 0$

Step 2: Factor out common powers of 5

Notice that both terms involve powers of 5. Let's express them as powers of 5 and try to factor. The equation is:

$2 \cdot 5^{2x+2} + 5^{x+2} - 3 = 0$

Since $5^{x+2}$ is a common term, let's factor it:

$5^{x+2}(2 \cdot 5^x + 1) - 3 = 0$

Step 3: Solve for $x$

At this point, solving the equation directly might involve trial and error, logarithmic functions, or checking the possible values of $x$.

By testing potential values, or solving further, we