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Let's go through the factorization of each quadratic expression one by one:

### o. \( 3x^2 + 16x + 5 \)
We will factor by splitting the middle term:

1. Multiply \( 3 \times 5 = 15 \).
2. Find two numbers that multiply to 15 and add to 16: these numbers are 15 and 1.
3. Split the middle term: 
   \[
   3x^2 + 15x + x + 5
   \]
4. Factor by grouping:
   \[
   3x(x + 5) + 1(x + 5)
   \]
5. Final factorization:
   \[
   (3x + 1)(x + 5)
   \]

### p. \( 3x^2 + 14x + 8 \)
1. Multiply \( 3 \times 8 = 24 \).
2. Find two numbers that multiply to 24 and add to 14: these numbers are 12 and 2.
3. Split the middle term:
   \[
   3x^2 + 12x + 2x + 8
   \]
4. Factor by grouping:
   \[
   3x(x + 4) + 2(x + 4)
   \]
5. Final factorization:
   \[
   (3x + 2)(x + 4)
   \]

### q. \( 3x^2 + 8x + 5 \)
1. Multiply \( 3 \times 5 = 15 \).
2. Find two numbers that multiply to 15 and add to 8: these numbers are 3 and 5.
3. Split the middle term:
   \[
   3x^2 + 3x + 5x + 5
   \]
4. Factor by grouping:
   \[
   3x(x + 1) + 5(x + 1)
   \]
5. Final factorization:
   \[
   (3x + 5)(x + 1)
   \]

### r. \( 4x^2 + 7x + 3 \)
1. Multiply \( 4 \times 3 = 12 \).
2. Find two numbers that multiply to 12 and add to 7: these numbers are 3 and 4.
3. Split the middle term:
   \[
   4x^2 + 4x + 3x + 3
   \]
4. Factor by grouping:
   \[
   4x(x + 1) + 3(x + 1)
   \]
5. Final factorization:
   \[
   (4x + 3)(x + 1)
   \]

### s. \( 4x^2 + 13x + 3 \)
1. Multiply \( 4 \times 3 = 12 \).
2. Find two numbers that multiply to 12 and add to 13: these numbers are 12 and 1.
3. Split the middle term:
   \[
   4x^2 + 12x + x + 3
   \]
4. Factor by grouping:
   \[
   4x(x + 3) + 1(x + 3)
   \]
5. Final factorization:
   \[
   (4x + 1)(x + 3)
   \]

### t. \( 4x^2 + 11x + 6 \)
1. Multiply \( 4 \times 6 = 24 \).
2. Find two numbers that multiply to 24 and add to 11: these numbers are 8 and 3.
3. Split the middle term:
   \[
   4x^2 + 8x + 3x + 6
   \]
4. Factor by grouping:
   \[
   4x(x + 2) + 3(x + 2)
   \]
5. Final factorization:
   \[
   (4x + 3)(x + 2)
   \]

### u. \( 5x^2 - 7x - 6 \)
1. Multiply \( 5 \times (-6) = -30 \).
2. Find two numbers that multiply to -30 and add to -7: these numbers are -10 and 3.
3. Split the middle term:
   \[
   5x^2 - 10x + 3x - 6
   \]
4. Factor by grouping:
   \[
   5x(x - 2) + 3(x - 2)
   \]
5. Final factorization:
   \[
   (5x + 3)(x - 2)
   \]

### v. \( 3x^2 - 13x - 10 \)
1. Multiply \( 3 \times (-10) = -30 \).
2. Find two numbers that multiply to -30 and add to -13: these numbers are -15 and 2.
3. Split the middle term:
   \[
   3x^2 - 15x + 2x - 10
   \]
4. Factor by grouping:
   \[
   3x(x - 5) + 2(x - 5)
   \]
5. Final factorization:
   \[
   (3x + 2)(x - 5)
   \]

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Do you have any questions, or would you like more details on any step? Here are some additional questions related to factorization:

1. How do we factor trinomials where the leading coefficient is greater than 1?
2. What strategies can be used when factoring expressions where factoring by grouping doesn't work?
3. Can all quadratics be factored using integers? If not, what should be done?
4. How does factoring relate to finding the roots of a quadratic equation?
5. What is the difference between factoring and expanding an expression?

**Tip**: Always double-check your factorization by expanding the factors back to the original quadratic to ensure it's correct!

Factor all of these quadratic equations: 3x^2 + 16x + 5, 3x^2 + 14x + 8, 3x^2 + 8x + 5, 4x^2 + 7x + 3, 4x^2 + 13x + 3, 4x^2 + 11x + 6, 5x^2 - 7x - 6, 3x^2 - 13x - 10

Learn how to factor quadratic equations step-by-step. This lesson covers factorization of quadratic expressions such as 3x^2 + 16x + 5, 4x^2 + 7x + 3, and others. Suitable for students in Grades 8-10.

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