https://file.aiquickdraw.com/mathbot/file/1726601631056umo98z19.jpg

The image contains four quadratic expressions, which I will now factor where possible:

a) \( x^2 + 16x + 63 \)

b) \( x^2 - 11x + 28 \)

c) \( x^2 - x - 56 \)

d) \( x^2 + x - 6 \)

### Factorization:

1. **(a)** \( x^2 + 16x + 63 \)
   - We are looking for two numbers that multiply to 63 and add to 16. The numbers are 7 and 9.
   - Factorization: \( (x + 7)(x + 9) \).

2. **(b)** \( x^2 - 11x + 28 \)
   - We are looking for two numbers that multiply to 28 and add to -11. The numbers are -4 and -7.
   - Factorization: \( (x - 4)(x - 7) \).

3. **(c)** \( x^2 - x - 56 \)
   - We are looking for two numbers that multiply to -56 and add to -1. The numbers are -8 and 7.
   - Factorization: \( (x - 8)(x + 7) \).

4. **(d)** \( x^2 + x - 6 \)
   - We are looking for two numbers that multiply to -6 and add to 1. The numbers are 3 and -2.
   - Factorization: \( (x + 3)(x - 2) \).

### Summary of Factored Forms:
- **(a)** \( (x + 7)(x + 9) \)
- **(b)** \( (x - 4)(x - 7) \)
- **(c)** \( (x - 8)(x + 7) \)
- **(d)** \( (x + 3)(x - 2) \)

Would you like more details or explanations on any of the factorization steps?

Here are 5 related questions for further practice:
1. How would you solve for the roots of each quadratic equation?
2. What is the sum and product of the roots for the quadratic \( x^2 + 16x + 63 \)?
3. Can you explain how factoring helps in solving quadratic equations?
4. How would you solve \( x^2 + 7x + 12 = 0 \) by factoring?
5. What are the conditions required for a quadratic to be factorable?

**Tip:** Always check your factorization by expanding the factors back to the original quadratic expression to ensure correctness.

Factor the following quadratic expressions:

a) x^2 + 16x + 63
b) x^2 - 11x + 28
c) x^2 - x - 56
d) x^2 + x - 6

Learn how to factor quadratic expressions like x^2 + 16x + 63, x^2 - 11x + 28, x^2 - x - 56, and x^2 + x - 6. This guide provides detailed steps to factor each expression using algebraic techniques suitable for students in Grades 8-10.

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