Let's solve the given equation:

\[
4x^2 - 25y^2 - 3x + 164 = 0
\]

### Step 1: Group similar terms involving \( x \) and \( y \).
We can rewrite the equation as:

\[
4x^2 - 3x - 25y^2 = -164
\]

### Step 2: Complete the square for the \( x \)-terms.
Take the \( x \)-terms \( 4x^2 - 3x \). First, factor out the coefficient of \( x^2 \), which is 4:

\[
4(x^2 - \frac{3}{4}x)
\]

Now, complete the square inside the parentheses. To do this, take half of \( \frac{3}{4} \), square it, and add/subtract inside the square:

\[
x^2 - \frac{3}{4}x + \left(\frac{3}{8}\right)^2 - \left(\frac{3}{8}\right)^2 = x^2 - \frac{3}{4}x + \frac{9}{64} - \frac{9}{64}
\]

Now, we can factor the completed square:

\[
4\left(\left(x - \frac{3}{8}\right)^2 - \frac{9}{64}\right)
\]

Simplify the term:

\[
4\left(x - \frac{3}{8}\right)^2 - \frac{36}{64} = 4\left(x - \frac{3}{8}\right)^2 - \frac{9}{16}
\]

### Step 3: Substitute this into the original equation.

Substituting this into the equation gives:

\[
4\left(x - \frac{3}{8}\right)^2 - \frac{9}{16} - 25y^2 = -164
\]

### Step 4: Simplify the equation.

Move all constant terms to the right-hand side:

\[
4\left(x - \frac{3}{8}\right)^2 - 25y^2 = -164 + \frac{9}{16}
\]

The right-hand side becomes:

\[
-164 + \frac{9}{16} = \frac{-10496 + 9}{16} = \frac{-10487}{16}
\]

Thus, the equation becomes:

\[
4\left(x - \frac{3}{8}\right)^2 - 25y^2 = \frac{-10487}{16}
\]

This is the simplified form of the equation.

Would you like more details on a specific part of the solution, or do you have any further questions?

### Related questions:
1. How do you complete the square for different types of equations?
2. What are the steps for solving systems of nonlinear equations?
3. How do conic sections relate to equations like this?
4. Can this equation represent a hyperbola? Why or why not?
5. What methods are used to classify second-degree equations?

### Tip:
When completing the square, always remember to balance both sides of the equation by adding or subtracting the same value.

This problem involves solving the equation 4x^2 - 25y^2 - 3x + 164 = 0 by completing the square. It covers key algebraic concepts, including conic sections and quadratic equations, making it suitable for high school students in Grades 10-12.

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