The expression you've provided is:

$\frac{x^2}{x + 7} - \frac{49}{x + 7}$

Let's simplify this step by step.

Step 1: Factor the numerator of the first term

Notice that the second term, $\frac{49}{x + 7}$, has the same denominator as the first term. So, we can combine these terms into a single fraction.

$\frac{x^2}{x + 7} - \frac{49}{x + 7} = \frac{x^2 - 49}{x + 7}$

Step 2: Factor the numerator

The expression $x^2 - 49$ is a difference of squares. It can be factored as:

$x^2 - 49 = (x + 7)(x - 7)$

Now, substitute this back into the fraction:

$\frac{(x + 7)(x - 7)}{x + 7}$

Step 3: Cancel common terms

You can cancel out the common factor of $(x + 7)$ from the numerator and the denominator (note that this cancellation is valid only when $x \neq -7$ because division by zero is undefined):

$x - 7$

Final Result:

The simplified expression is:

$x - 7 \quad \text{(for } x \neq -7\text{)}$

Would you like more details or have any questions on this simplification?

Here are 5 related questions:

1. What happens when $x = -7$ in the original expression?
2. How do you handle expressions with higher powers, such as $\frac{x^3}{x+7}$?
3. What is the difference between a sum and difference of squares in algebra?
4. Can you simplify $\frac{4x^2 - 16}{x + 4}$ using the same method?
5. What is the importance of factoring in simplifying rational expressions?

Tip: Always check for restrictions in rational expressions. In this case, $x = -7$ is a restriction because it would make the denominator zero.