The provided problem asks for the indicated operations on the following three expressions:

$x + \frac{4}{x^2 - 16} - \frac{x^2}{x + 4}$

We are required to simplify this expression and express the answer in the form $\frac{A}{B}$.

Let's begin solving this.

Step 1: Factor the denominator of the second expression.

The denominator $x^2 - 16$ is a difference of squares: $x^2 - 16 = (x - 4)(x + 4)$ Thus, the second term becomes: $\frac{4}{(x - 4)(x + 4)}$

Step 2: Simplify the third expression.

The third term is $\frac{x^2}{x + 4}$. We leave this as is for now.

Step 3: Combine all terms under a common denominator.

To combine the expressions, we need a common denominator. The common denominator will be $(x - 4)(x + 4)$.

Let's rewrite the terms with this common denominator:

• The first term $x$ can be written as $\frac{x(x - 4)(x + 4)}{(x - 4)(x + 4)} = \frac{x(x^2 - 16)}{(x - 4)(x + 4)}$.
• The second term is already in the desired form: $\frac{4}{(x - 4)(x + 4)}$.
• The third term $\frac{x^2}{x + 4}$ can be rewritten as $\frac{x^2(x - 4)}{(x - 4)(x + 4)}$.

Now, let's combine the terms and simplify.The simplified expression is:

$\frac{4(x^2 - 4x + 1)}{x^2 - 16}$

This is in the form $\frac{A}{B}$, where:

$A = 4(x^2 - 4x + 1)$ and $B = x^2 - 16$

Thus:

$A = 4(x^2 - 4x + 1) = 4x^2 - 16x + 4$ $B = (x - 4)(x + 4)$

Final Answer:

• $A = 4x^2 - 16x + 4$
• $B = x^2 - 16$

Would you like more details on any of the steps? Here are five related questions for further exploration:

1. How do you factor a quadratic expression like $x^2 - 16$?
2. What are the steps to combine fractions with different denominators?
3. How can you apply the difference of squares to simplify expressions?
4. Why is it important to find a common denominator when adding or subtracting fractions?
5. How do you expand and simplify polynomials like $4(x^2 - 4x + 1)$?

Tip: Always check if the final expression can be factored further to simplify it even more.