The expressions in the image are:

6. $7k^2 + 9k$
7. $2b^2 + 17b + 21$
8. $28n^4 + 16n^3 - 80n^2$

Let’s factor each expression.

Problem 6: $7k^2 + 9k$

1. Identify common factors:

   • Both terms contain $k$, so we can factor out $k$.

2. Factor the expression: $7k^2 + 9k = k(7k + 9)$

Factored form: $k(7k + 9)$

Problem 8: $2b^2 + 17b + 21$

1. Look for factors of $2 \times 21 = 42$ that add up to 17:

   • The factors 3 and 14 work because $3 + 14 = 17$.

2. Rewrite and factor by grouping: $2b^2 + 17b + 21 = 2b^2 + 14b + 3b + 21$ $= 2b(b + 7) + 3(b + 7)$ $= (2b + 3)(b + 7)$

Factored form: $(2b + 3)(b + 7)$

Problem 10: $28n^4 + 16n^3 - 80n^2$

1. Identify the common factor:

   • All terms are divisible by $4n^2$.

2. Factor out $4n^2$: $28n^4 + 16n^3 - 80n^2 = 4n^2(7n^2 + 4n - 20)$

3. Factor the quadratic $7n^2 + 4n - 20$:

   • Look for factors of $7 \times -20 = -140$ that add up to 4: 14 and -10 work because $14 - 10 = 4$.

4. Rewrite and factor by grouping: $4n^2(7n^2 + 4n - 20) = 4n^2(7n^2 + 14n - 10n - 20)$ $= 4n^2(7n(n + 2) - 10(n + 2))$ $= 4n^2(7n - 10)(n + 2)$

Factored form: $4n^2(7n - 10)(n + 2)$

Would you like further clarification on any of these solutions, or more details?

Related Questions:

1. How do you determine the greatest common factor in polynomial expressions?
2. What is the process of factoring by grouping?
3. How do you handle factoring when the leading coefficient is greater than 1?
4. What strategies are useful for recognizing patterns in polynomials?
5. How can you check if your factored solution is correct?

Tip:

When factoring trinomials, always check if the expression can be simplified further after factoring out the greatest common factor first.