https://file.aiquickdraw.com/mathbot/file/1726910001967aza1nitj.jpg

Let's solve both of the problems shown in the image:

### First Problem: Factoring \( 2y^2 + 12y - 54 \)

1. First, notice that 2 is a common factor for all terms:
   \[
   2(y^2 + 6y - 27)
   \]
2. Now, factor \( y^2 + 6y - 27 \). We need two numbers that multiply to \(-27\) and add to \(6\). These numbers are \(9\) and \(-3\).
   \[
   y^2 + 6y - 27 = (y + 9)(y - 3)
   \]
3. Finally, factor completely:
   \[
   2(y + 9)(y - 3)
   \]

Thus, the correct answer is: **\( 2(y + 9)(y - 3) \)**.

### Second Problem: Factoring \( 21xy^3 - 15x^2y^2 \)

1. We are given that one factor is \( 3xy^2 \).
2. Factor out \( 3xy^2 \) from \( 21xy^3 - 15x^2y^2 \):
   \[
   21xy^3 - 15x^2y^2 = 3xy^2(7y - 5x)
   \]

Thus, the other factor is: **\( 7y - 5x \)**.

Let me know if you'd like further details!

### Follow-up Questions:
1. How do you find the greatest common factor in a polynomial?
2. What does it mean for a polynomial to be factored completely?
3. How can you check if a factorization is correct?
4. What are common methods for factoring quadratic expressions?
5. What role do coefficients play in factoring?

### Tip:
When factoring, always start by checking for a greatest common factor before applying other methods like grouping or the quadratic formula.

Factoring quadratic expressions and polynomials.

Learn how to factor the quadratic expression 2y^2 + 12y - 54 and the polynomial 21xy^3 - 15x^2y^2 with detailed steps. This solution explores factoring by grouping and finding the greatest common factor.

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