Given \(a = \frac{7}{2}\), we need to solve \(gf(x) = 0\).

First, find \(f(x)\):

\(f(x) = \left(x + \frac{7}{2}\right)^2 - \frac{7}{2}\)

For \(x \leq -\frac{7}{2}\), \(f(x) = \left(x + \frac{7}{2}\right)^2 - \frac{7}{2}\).

Now, find \(gf(x)\):

\(gf(x) = g(f(x)) = 2f(x) - 1\)

Set \(gf(x) = 0\):

\(2f(x) - 1 = 0\)

\(2\left(\left(x + \frac{7}{2}\right)^2 - \frac{7}{2}\right) - 1 = 0\)

\(2\left(x + \frac{7}{2}\right)^2 - 7 - 1 = 0\)

\(2\left(x + \frac{7}{2}\right)^2 = 8\)

\(\left(x + \frac{7}{2}\right)^2 = 4\)

\(x + \frac{7}{2} = \pm 2\)

\(x = -\frac{7}{2} \pm 2\)

\(x = -\frac{7}{2} + 2 = -\frac{3}{2}\)

\(x = -\frac{7}{2} - 2 = -\frac{11}{2}\)

Since \(x \leq -\frac{7}{2}\), the valid solution is \(x = -\frac{11}{2}\).