https://file.aiquickdraw.com/mathbot/file/17258454651537r1m1e6m.jpg

Let's go through the problems in the image. There are three sections with different tasks:

### 1. Factor the following trinomials:
We are asked to factor the given trinomials.

- \( a. \ y^2 + 10y + 24 \)
- \( b. \ x^2 - x - 20 \)
- \( c. \ y^2 + 3y + 2 \)
- \( d. \ x^2 - 13x + 36 \)
- \( e. \ y^2 + 7y + 12 \)
- \( f. \ x^2 + 18x + 81 \)

### 2. Complete the expression with the term necessary to make it a perfect square trinomial:
Here, we need to complete the square for each given binomial.

- \( a. \ y^2 + \_ = x + 49 \)
- \( b. \ x^2 + \_ = x + 1 \)
- \( c. \ x^2 + 10x + \_ \)
- \( d. \ x^2 - x + \_ = x + 36 \)
- \( e. \ y^2 + 6y + \_ \)
- \( f. \ x^2 - 16x + \_ \)
- \( g. \ y^2 + 18y + \_ \)
- \( h. \ x^2 + x + \_ = x + 121 \)

### 3. Verify if the trinomials have been factored correctly; if not, identify the error and correct it:

- \( a. \ y^2 + 13y + 42 = (y + 7)(y + 6) \)
- \( b. \ x^2 - 10x + 25 = (y - 5)(y - 2) \)
- \( c. \ p^2 - 7p - 20 = (p - 5)(p + 4) \)
- \( d. \ y^2 - 10y + 21 = (y - 7)(y - 3) \)

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Factor the given trinomials, complete the expression for a perfect square trinomial, and verify if the trinomials have been factored correctly.

Practice factoring trinomials and completing the square to form perfect square trinomials. This problem set is suitable for Grades 8-10 students, covering key algebra concepts such as factoring, quadratic identities, and completing the square.

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