Simplify the expression.
StartFraction 4 Over x minus 7 EndFraction times StartFraction x squared minus 49 Over 8 EndFraction plus 6

We are tasked with simplifying the expression:

\[
\frac{4}{x - 7} \times \frac{x^2 - 49}{8} + 6
\]

### Step 1: Factor \(x^2 - 49\)
The term \(x^2 - 49\) is a difference of squares, which can be factored as:

\[
x^2 - 49 = (x - 7)(x + 7)
\]

Now the expression becomes:

\[
\frac{4}{x - 7} \times \frac{(x - 7)(x + 7)}{8} + 6
\]

### Step 2: Simplify the product
Notice that \(x - 7\) appears in both the numerator and denominator, so it cancels out:

\[
\frac{4}{x - 7} \times \frac{(x - 7)(x + 7)}{8} = \frac{4(x + 7)}{8}
\]

This simplifies to:

\[
\frac{(x + 7)}{2}
\]

### Step 3: Add 6
Now the expression becomes:

\[
\frac{x + 7}{2} + 6
\]

To combine these, express 6 as a fraction with denominator 2:

\[
6 = \frac{12}{2}
\]

Now we can add the fractions:

\[
\frac{x + 7}{2} + \frac{12}{2} = \frac{x + 7 + 12}{2} = \frac{x + 19}{2}
\]

### Final Answer:
The simplified expression is:

\[
\frac{x + 19}{2}
\]

Would you like more details on any of the steps? 

Here are 5 related questions you might find interesting:
1. How do you simplify a product of fractions?
2. What is the difference of squares, and how is it used in factoring?
3. How do you combine terms in an algebraic fraction?
4. What other methods can be used to simplify algebraic expressions?
5. Can expressions involving variables always be simplified further?

**Tip:** When simplifying fractions with algebraic terms, always look for common factors to cancel out.

Simplify the expression. StartFraction 4 Over x minus 7 EndFraction times StartFraction x squared minus 49 Over 8 EndFraction plus 6

Learn how to simplify the algebraic expression involving fractions, factoring, and difference of squares: (4/(x - 7)) * ((x^2 - 49)/8) + 6. This detailed solution demonstrates key algebraic concepts and techniques suitable for Grades 8-10.

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