https://file.aiquickdraw.com/mathbot/file/1726068594623er75odrs.jpg

From the image, we have two separate expressions written down, possibly two fractions that look like this:

\[
\frac{x^2 - 16}{x^2 - 10 + 25} \quad \text{and} \quad \frac{3x - 12}{x^2 - 3x - 10}
\]

Let's simplify each fraction one at a time:

### First Fraction: \(\frac{x^2 - 16}{x^2 - 10 + 25}\)

1. The numerator is \(x^2 - 16\), which is a difference of squares:
   \[
   x^2 - 16 = (x - 4)(x + 4)
   \]

2. The denominator is \(x^2 - 10 + 25 = x^2 + 15\).

So, the simplified first fraction is:
\[
\frac{(x - 4)(x + 4)}{x^2 + 15}
\]

### Second Fraction: \(\frac{3x - 12}{x^2 - 3x - 10}\)

1. The numerator is \(3x - 12\). This can be factored as:
   \[
   3x - 12 = 3(x - 4)
   \]

2. The denominator is \(x^2 - 3x - 10\). We need to factor this quadratic expression:
   \[
   x^2 - 3x - 10 = (x - 5)(x + 2)
   \]

Thus, the simplified second fraction is:
\[
\frac{3(x - 4)}{(x - 5)(x + 2)}
\]

### Final Simplified Fractions

- First fraction: \(\frac{(x - 4)(x + 4)}{x^2 + 15}\)
- Second fraction: \(\frac{3(x - 4)}{(x - 5)(x + 2)}\)

Would you like to explore any further steps or clarifications on these simplifications?

Here are 5 related questions:
1. How do you factor more complex quadratic expressions?
2. What is the importance of factoring in simplifying rational expressions?
3. How can we identify a difference of squares pattern?
4. What are the common techniques to simplify fractions in algebra?
5. How can factoring quadratics be applied to solving equations?

**Tip:** Always check if the quadratic can be factored easily using common techniques like grouping, difference of squares, or trial and error.

Simplify the following two algebraic fractions: (x^2 - 16) / (x^2 - 10 + 25) and (3x - 12) / (x^2 - 3x - 10).

Learn how to simplify the algebraic fractions (x^2 - 16) / (x^2 - 10 + 25) and (3x - 12) / (x^2 - 3x - 10) by factoring quadratic expressions and recognizing a difference of squares pattern. This step-by-step guide is ideal for students in Grades 8-10.

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