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To factor each quadratic expression, let's solve each problem step by step.

### 1. \( x^2 + 8x - 20 \)

We need to factor this expression into the form:
\[
(x + a)(x + b)
\]
where \( a \) and \( b \) are numbers that satisfy two conditions:
1. Their product is \(-20\).
2. Their sum is \(8\).

After testing factors of \(-20\) that add up to \(8\), we see that \(10\) and \(-2\) satisfy these conditions.

So, we can write:
\[
x^2 + 8x - 20 = (x + 10)(x - 2)
\]

### 2. \( x^2 - 2x - 48 \)

We want to factor this expression into:
\[
(x + a)(x + b)
\]
where \( a \) and \( b \) are numbers that satisfy:
1. Their product is \(-48\).
2. Their sum is \(-2\).

After examining the factors of \(-48\), we find that \(6\) and \(-8\) meet these conditions.

Thus, we can write:
\[
x^2 - 2x - 48 = (x + 6)(x - 8)
\]

### Final Answers:

1. \( x^2 + 8x - 20 = (x + 10)(x - 2) \)
2. \( x^2 - 2x - 48 = (x + 6)(x - 8) \)

Would you like more details on any part of this factoring process, or have any questions?

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Here are five related questions to deepen understanding:

1. How do you identify the factors that add up to a given number?
2. Why does changing the sign of one factor affect the sum and product?
3. Can all quadratics be factored using integers? Why or why not?
4. What are alternative methods if factoring by inspection is challenging?
5. How does factoring help in solving quadratic equations?

**Tip:** Remember, if a quadratic expression doesn't factor easily, you can use the quadratic formula to find its roots.

Factor the following quadratic expressions:
1. x^2 + 8x - 20
2. x^2 - 2x - 48

Learn how to factor the quadratic expressions x^2 + 8x - 20 and x^2 - 2x - 48. This solution demonstrates the process of finding pairs of factors that satisfy the sum and product conditions in quadratic equations. Ideal for students in Grades 8-10 studying factoring and algebra.

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