To solve the equation \((8 - x - 1)^{2x - 1} = 5^{2x + 1}\), let's first simplify the expression inside the parentheses on the left-hand side.

\[
8 - x - 1 = 7 - x
\]

So the equation becomes:

\[
(7 - x)^{2x - 1} = 5^{2x + 1}
\]

To solve this equation, we need to find \(x\) such that both sides are equal. One approach is to see if there is an obvious solution by inspection.

Let's try \(x = 2\):

\[
(7 - 2)^{2 \cdot 2 - 1} = 5^{2 \cdot 2 + 1}
\]

Simplify both sides:

\[
5^{4 - 1} = 5^{4 + 1}
\]

\[
5^3 = 5^5
\]

This is not true, so \(x = 2\) is not a solution.

Next, let's try \(x = 1\):

\[
(7 - 1)^{2 \cdot 1 - 1} = 5^{2 \cdot 1 + 1}
\]

Simplify both sides:

\[
6^{2 - 1} = 5^{2 + 1}
\]

\[
6^1 = 5^3
\]

This is also not true, so \(x = 1\) is not a solution.

Next, let's try \(x = 0\):

\[
(7 - 0)^{2 \cdot 0 - 1} = 5^{2 \cdot 0 + 1}
\]

Simplify both sides:

\[
7^{-1} = 5^1
\]

\[
\frac{1}{7} = 5
\]

This is not true, so \(x = 0\) is not a solution.

Since we are not finding an obvious solution by inspection, we might need to use logarithms or numerical methods to solve this equation.

Taking the natural logarithm (ln) of both sides:

\[
\ln((7 - x)^{2x - 1}) = \ln(5^{2x + 1})
\]

Using the property of logarithms \(\ln(a^b) = b \ln(a)\):

\[
(2x - 1) \ln(7 - x) = (2x + 1) \ln(5)
\]

This is a transcendental equation and generally requires numerical methods to solve. One approach is to use a numerical solver or graphing tool to find the intersection of the functions:

\[
f(x) = (2x - 1) \ln(7 - x)
\]

and

\[
g(x) = (2x + 1) \ln(5)
\]

By plotting these functions or using a numerical solver, we can find the value of \(x\) that satisfies the equation.