It appears you’ve uploaded an image of a homework sheet focused on factoring polynomials and trinomials. Let's go through the problems shown and provide factoring steps.

The sheet contains two main sections:

1. Factor each polynomial. Check your answer by distributing.
2. Factor each polynomial, looking for a Greatest Common Factor (GCF) first.

I'll start by addressing a few examples to illustrate the process. Please let me know if you need more details on each or specific problems solved.

Section 1: Factor each polynomial

1. $2x^2 + 5x + 2$

   • To factor, look for two numbers that multiply to $2 \times 2 = 4$ and add up to $5$.
   • The numbers $4$ and $1$ work here.
   • So, rewrite as $2x^2 + 4x + x + 2$.
   • Factor by grouping: $2x(x + 2) + 1(x + 2)$.
   • Final answer: $(2x + 1)(x + 2)$.

2. $x^2 + 9x - 5$

   • This trinomial does not factor nicely using integers (as there are no integer factors of $-5$ that add to $9$).
   • This could be left as is, or solved using the quadratic formula if needed.

3. $2x^2 + 9x - 5$

   • Find two numbers that multiply to $-10$ (from $2 \times -5$) and add up to $9$.
   • The numbers $10$ and $-1$ work.
   • Rewrite as $2x^2 + 10x - x - 5$.
   • Factor by grouping: $2x(x + 5) - 1(x + 5)$.
   • Final answer: $(2x - 1)(x + 5)$.

Section 2: Factor each polynomial, looking for a GCF

10. $6x^3 + 9x - 27$
    • Start by factoring out the GCF, which is $3$: $3(2x^3 + 3x - 9)$.
    • After factoring out the GCF, examine the remaining polynomial to see if further factoring is possible.

Would you like detailed solutions for each problem on the sheet, or do you need specific problems?

Here are some related questions for deeper practice:

1. How can we identify when a trinomial is factorable using integers?
2. Why is it helpful to factor out the GCF before attempting other factoring techniques?
3. What is the purpose of checking the factors by distributing?
4. Can factoring be applied to solve quadratic equations?
5. How does grouping work in factoring trinomials?

Tip: Always start by factoring out any common factors first, as this simplifies the remaining expression and makes further factoring easier.