Start by using the property \(\log_{10} 10 = 1\) and \(\log_{10} 10^{-1} = -1\).

Rearrange the equation:

\(\log_{10}(2x + 1) = 2\log_{10}(x + 1) - 1\)

\(\log_{10}(2x + 1) + 1 = 2\log_{10}(x + 1)\)

\(\log_{10}(10(2x + 1)) = \log_{10}((x + 1)^2)\)

Equate the arguments:

\(10(2x + 1) = (x + 1)^2\)

Expand and simplify:

\(20x + 10 = x^2 + 2x + 1\)

\(x^2 - 18x - 9 = 0\)

Solve the quadratic equation using the quadratic formula:

\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

where \(a = 1, b = -18, c = -9\).

\(x = \frac{18 \pm \sqrt{324 + 36}}{2}\)

\(x = \frac{18 \pm \sqrt{360}}{2}\)

\(x = \frac{18 \pm 18.9737}{2}\)

\(x = 18.487\) and \(x = -0.487\)