We are tasked with solving the expression:

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

### Step 1: Expand the product \(\left(\frac{x}{4} + 4\right)\left(x - \frac{1}{4}\right)\)

We'll use the distributive property (FOIL method) to expand:

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

Simplifying each term:

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

Now combining these, we get:

\[
\frac{x^2}{4} - \frac{x}{16} + 4x - 1
\]

### Step 2: Subtract \( (x^2 - 1) \) from the result

Now we subtract \( (x^2 - 1) \) from the above expression:

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

Simplifying this expression:

\[
= \frac{x^2}{4} - \frac{x}{16} + 4x - 1 - x^2 + 1
\]

### Step 3: Combine like terms

Now combine the terms:

\[
\frac{x^2}{4} - x^2 + 4x - \frac{x}{16} - 1 + 1
\]

First, simplify the \(x^2\) terms:

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

The constants \(-1 + 1\) cancel each other out, so we are left with:

\[
-\frac{3x^2}{4} + 4x - \frac{x}{16}
\]

### Step 4: Combine the linear terms

To combine \(4x\) and \(-\frac{x}{16}\), we need a common denominator:

\[
4x = \frac{64x}{16}
\]

So,

\[
\frac{64x}{16} - \frac{x}{16} = \frac{63x}{16}
\]

### Final expression

Thus, the simplified expression is:

\[
-\frac{3x^2}{4} + \frac{63x}{16}
\]

This is the final result.

---

Would you like more details on any of these steps? Here are some questions to explore the topic further:

1. How do you simplify algebraic expressions with fractions?
2. Can you explain the FOIL method in more detail?
3. What are some alternative ways to expand binomials?
4. How do you solve quadratic equations after simplifying them?
5. How do you factor quadratic expressions?

**Tip:** Always combine like terms to simplify expressions as much as possible before proceeding to more complex operations.

Learn how to simplify the algebraic expression (x/4 + 4)(x - 1/4) - (x² - 1) step by step. This solution covers the use of binomial expansion, distributive property, and combining like terms. Suitable for students in Grades 9-11.

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