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Let's go through each of the problems step by step and factor the given polynomials.

### Problem 1: Write the following polynomials as a product of factors.

1. \( x^2 + 10x + 25 \)

This is a perfect square trinomial. We can factor it as follows:

\[
x^2 + 10x + 25 = (x + 5)^2
\]

2. \( x^2 - 20x + 100 \)

This is also a perfect square trinomial:

\[
x^2 - 20x + 100 = (x - 10)^2
\]

### Problem 2: Factor the following polynomials.

1. \( 9x^2 - 24x + 16 \)

This is a perfect square trinomial of the form \( (ax - b)^2 \):

\[
9x^2 - 24x + 16 = (3x - 4)^2
\]

2. \( 4x^2 - x - 14 \)

This polynomial does not appear to be a perfect square, so we need to factor it using the quadratic formula or factorization techniques. First, we'll try to factor it directly by finding two numbers that multiply to \( 4 \times -14 = -56 \) and add up to \( -1 \). These numbers are \( -8 \) and \( 7 \), so we can factor it as follows:

\[
4x^2 - x - 14 = (4x + 7)(x - 2)
\]

### Summary:
1. \( x^2 + 10x + 25 = (x + 5)^2 \)
2. \( x^2 - 20x + 100 = (x - 10)^2 \)
3. \( 9x^2 - 24x + 16 = (3x - 4)^2 \)
4. \( 4x^2 - x - 14 = (4x + 7)(x - 2) \)

Let me know if you'd like more details on any of these factorizations!

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Here are five related questions for you to practice:
1. Factor \( x^2 - 6x + 9 \).
2. Factor \( 16x^2 + 24x + 9 \).
3. Factor \( x^2 + 12x + 36 \).
4. Factor \( 6x^2 - 19x + 10 \).
5. Factor \( 2x^2 - 8x + 6 \).

### Tip:
When factoring, always check if the quadratic is a perfect square trinomial, which can save time by applying the formula \( (ax + b)^2 \).

Write the following polynomials as a product of factors: 

1. x^2 + 10x + 25 
2. x^2 - 20x + 100 

Factor the following polynomials: 

1. 9x^2 - 24x + 16 
2. 4x^2 - x - 14

Learn how to factor polynomials such as x^2 + 10x + 25, x^2 - 20x + 100, 9x^2 - 24x + 16, and 4x^2 - x - 14. This guide covers perfect square trinomials and the use of the quadratic formula for factorization. Suitable for students in Grades 8-10.

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