To find the value of \(x^4 + \frac{1}{x^4}\) given that \(x + \frac{1}{x} = 4\), follow these steps:

1. Square both sides of \(x + \frac{1}{x} = 4\):

\[
\left(x + \frac{1}{x}\right)^2 = 4^2
\]
\[
x^2 + 2 \cdot \frac{x}{x} + \frac{1}{x^2} = 16
\]
\[
x^2 + 2 + \frac{1}{x^2} = 16
\]
\[
x^2 + \frac{1}{x^2} = 16 - 2
\]
\[
x^2 + \frac{1}{x^2} = 14
\]

2. Square \(x^2 + \frac{1}{x^2}\) to find \(x^4 + \frac{1}{x^4}\):

\[
\left(x^2 + \frac{1}{x^2}\right)^2 = 14^2
\]
\[
x^4 + 2 \cdot x^2 \cdot \frac{1}{x^2} + \frac{1}{x^4} = 196
\]
\[
x^4 + 2 + \frac{1}{x^4} = 196
\]
\[
x^4 + \frac{1}{x^4} = 196 - 2
\]
\[
x^4 + \frac{1}{x^4} = 194
\]

So, the value of \(x^4 + \frac{1}{x^4}\) is:

A. 194