Let's simplify the expression given in the image step by step. The expression appears to be:

$\frac{\frac{16a}{b} - \frac{b}{a}}{\frac{1}{a} - \frac{4}{b}}$

Step 1: Simplify the Numerator

The numerator is $\frac{16a}{b} - \frac{b}{a}$.

To subtract these fractions, we need a common denominator. The common denominator between $b$ and $a$ is $ab$. So we rewrite each fraction:

$\frac{16a}{b} = \frac{16a^2}{ab}, \quad \frac{b}{a} = \frac{b^2}{ab}$

Now, subtract them:

$\frac{16a^2}{ab} - \frac{b^2}{ab} = \frac{16a^2 - b^2}{ab}$

Step 2: Simplify the Denominator

The denominator is $\frac{1}{a} - \frac{4}{b}$.

Again, we need a common denominator, which is $ab$. Rewriting each fraction:

$\frac{1}{a} = \frac{b}{ab}, \quad \frac{4}{b} = \frac{4a}{ab}$

Now, subtract them:

$\frac{b}{ab} - \frac{4a}{ab} = \frac{b - 4a}{ab}$

Step 3: Form the Overall Fraction

Now we have the expression:

$\frac{\frac{16a^2 - b^2}{ab}}{\frac{b - 4a}{ab}}$

When dividing fractions, we multiply by the reciprocal of the denominator. So we get:

$\frac{16a^2 - b^2}{ab} \times \frac{ab}{b - 4a}$

The $ab$ terms cancel out, leaving us with:

$\frac{16a^2 - b^2}{b - 4a}$

Step 4: Factor the Numerator

The numerator $16a^2 - b^2$ is a difference of squares, which can be factored as:

$16a^2 - b^2 = (4a - b)(4a + b)$

Final Expression

Now we have:

$\frac{(4a - b)(4a + b)}{b - 4a}$

Notice that $b - 4a$ is the same as $-(4a - b)$. So, we can cancel out the $(4a - b)$ terms, but we are left with a negative sign:

$-(4a + b)$

Thus, the simplified expression is:

$-(4a + b)$

Final Answer:

$-(4a + b)$

Would you like further details or have any questions?

Here are five questions for further exploration:

1. What is the importance of finding a common denominator in fraction operations?
2. How can you identify the difference of squares when simplifying expressions?
3. Why does canceling terms in a rational expression require factoring first?
4. What are the conditions for canceling terms in algebraic fractions?
5. How does the negative sign affect the result when simplifying fractions?

Tip: Always check if terms in the numerator or denominator can be factored, as it often simplifies the process of solving.