Let's go ahead and solve the factoring problems you've shared:

1) Factor $3x^2 + 10x + 8$

We are looking for two numbers that multiply to $3 \times 8 = 24$ and add up to 10. These numbers are 4 and 6.

Now, split the middle term using these two numbers: $3x^2 + 4x + 6x + 8$ Group terms: $(3x^2 + 4x) + (6x + 8)$ Factor out the greatest common factor (GCF) from each group: $x(3x + 4) + 2(3x + 4)$ Now factor out the common binomial factor: $(3x + 4)(x + 2)$

2) Factor $4x^2 + 16x + 15$

We need two numbers that multiply to $4 \times 15 = 60$ and add up to 16. These numbers are 10 and 6.

Now, split the middle term: $4x^2 + 10x + 6x + 15$ Group terms: $(4x^2 + 10x) + (6x + 15)$ Factor out the GCF from each group: $2x(2x + 5) + 3(2x + 5)$ Factor out the common binomial factor: $(2x + 5)(2x + 3)$

3) Factor $2x^2 - 3x - 9$

We need two numbers that multiply to $2 \times (-9) = -18$ and add up to -3. These numbers are -6 and 3.

Now, split the middle term: $2x^2 - 6x + 3x - 9$ Group terms: $(2x^2 - 6x) + (3x - 9)$ Factor out the GCF from each group: $2x(x - 3) + 3(x - 3)$ Factor out the common binomial factor: $(2x + 3)(x - 3)$

4) Factor $3x^2 - 2x - 5$

We need two numbers that multiply to $3 \times (-5) = -15$ and add up to -2. These numbers are -5 and 3.

Now, split the middle term: $3x^2 - 5x + 3x - 5$ Group terms: $(3x^2 - 5x) + (3x - 5)$ Factor out the GCF from each group: $x(3x - 5) + 1(3x - 5)$ Factor out the common binomial factor: $(3x - 5)(x + 1)$

5) Factor $6x^2 - 13x + 6$

We need two numbers that multiply to $6 \times 6 = 36$ and add up to -13. These numbers are -9 and -4.

Now, split the middle term: $6x^2 - 9x - 4x + 6$ Group terms: $(6x^2 - 9x) + (-4x + 6)$ Factor out the GCF from each group: $3x(2x - 3) - 2(2x - 3)$ Factor out the common binomial factor: $(2x - 3)(3x - 2)$

These are the factored forms of all the expressions!

Would you like more details on any specific step, or do you have further questions? Here are 5 related questions to explore:

1. What is the difference between factoring using grouping and using the AC method?
2. How can you check if your factoring is correct after performing it?
3. Can all quadratic equations be factored, or are there cases where factoring isn’t possible?
4. What is the significance of the greatest common factor in the factoring process?
5. How do you factor higher-degree polynomials like cubic or quartic polynomials?

Tip: Always check your factored expression by expanding it back out to ensure it matches the original equation!