Let's work through the table and complete the missing columns.

Table Completion:

1. Given:
   $x^2 + x - 6$ and $x^2 - 9$
   Factored Form:
   $(x - 2)(x + 3)$ and $(x - 3)(x + 3)$
   Let Go:
   $x + 3$
   Be Unique:
   $x - 2$ and $x - 3$

2. Given:
   $15a$ and $12a^2b$
   Factored Form:
   $15a = 3 \times 5 \times a$
   $12a^2b = 3 \times 4 \times a \times ab$
   Let Go:
   $3a$
   Be Unique:
   $5$ and $4ab$

3. Given:
   $3x^2 - 12x$ and $6x^2 + 3x$
   Factored Form:
   $3x(x - 4)$ and $3x(2x + 1)$
   Let Go:
   $3x$
   Be Unique:
   $x - 4$ and $2x + 1$

4. Given:
   $x^2 + 4x + 3$ and $x^2 - 3x - 4$
   Factored Form:
   $(x + 1)(x + 3)$ and $(x - 4)(x + 1)$
   Let Go:
   $x + 1$
   Be Unique:
   $x + 3$ and $x - 4$

5. Given:
   $2x^2 + 11x + 5$ and $x^2 + 6x + 5$
   Factored Form:
   $(2x + 1)(x + 5)$ and $(x + 1)(x + 5)$
   Let Go:
   $x + 5$
   Be Unique:
   $2x + 1$ and $x + 1$

Guide Questions:

1. What techniques did you use to identify the factors of the given polynomials?

   • Factoring techniques, including finding the greatest common factor (GCF), factoring by grouping, and using formulas like the difference of squares or the quadratic formula.

2. If you are going to write the remaining factors in the Be Unique column as rational expressions, are these rational expressions in simplest form? Why or why not?

   • Yes, the remaining factors are in their simplest form since they have no common factors other than 1. Simplifying further wouldn't be possible.

Relative Questions:

1. What is the significance of finding the "Let Go" factors in such problems?
2. How does factoring relate to solving quadratic equations?
3. What are the differences between factoring simple quadratics and more complex polynomials?
4. Why is it important to ensure that expressions are in their simplest form when factoring?
5. Can you apply these factoring techniques to polynomials with more than two terms?

Tip:

When factoring, always check for the greatest common factor (GCF) first—it can simplify the process significantly.