Let’s go through the factorization of each quadratic expression. I'll show the step-by-step process for factoring each expression.

1. $p = x^2 - 9x + 20$

To factor this, we look for two numbers that multiply to $20$ and add up to $-9$:

• These numbers are $-4$ and $-5$.

So, $p = (x - 4)(x - 5)$

2. $s = x^2 + 4x - 12$

We need two numbers that multiply to $-12$ and add up to $4$:

• These numbers are $6$ and $-2$.

Thus, $s = (x + 6)(x - 2)$

3. $v = x^2 - 10x - 24$

We look for two numbers that multiply to $-24$ and add up to $-10$:

• These numbers are $-12$ and $2$.

Thus, $v = (x - 12)(x + 2)$

4. $q = x^2 + 2x - 3$

Here, we need numbers that multiply to $-3$ and add up to $2$:

• These numbers are $3$ and $-1$.

So, $q = (x + 3)(x - 1)$

5. $r = x^2 + x - 12$

For this, we need numbers that multiply to $-12$ and add up to $1$:

• These numbers are $4$ and $-3$.

Therefore, $r = (x + 4)(x - 3)$

6. $t = x^2 + 7x - 30$

We need two numbers that multiply to $-30$ and add up to $7$:

• These numbers are $10$ and $-3$.

Thus, $t = (x + 10)(x - 3)$

7. $w = x^2 - 7x - 30$

We look for two numbers that multiply to $-30$ and add up to $-7$:

• These numbers are $-10$ and $3$.

So, $w = (x - 10)(x + 3)$

8. $u = x^2 - x - 2$

For this, we need numbers that multiply to $-2$ and add up to $-1$:

• These numbers are $-2$ and $1$.

Thus, $u = (x - 2)(x + 1)$

9. $x = x^2 - x - 56$

Finally, we look for numbers that multiply to $-56$ and add up to $-1$:

• These numbers are $-8$ and $7$.

So, $x = (x - 8)(x + 7)$

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Summary of Factorizations:

1. $p = (x - 4)(x - 5)$
2. $s = (x + 6)(x - 2)$
3. $v = (x - 12)(x + 2)$
4. $q = (x + 3)(x - 1)$
5. $r = (x + 4)(x - 3)$
6. $t = (x + 10)(x - 3)$
7. $w = (x - 10)(x + 3)$
8. $u = (x - 2)(x + 1)$
9. $x = (x - 8)(x + 7)$

Let me know if you'd like more details on any specific step or if you have any questions!

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Follow-up Questions

1. How do you find factors of a quadratic expression quickly?
2. What methods other than factoring can be used to solve quadratic equations?
3. Why do some quadratic expressions not factor into real numbers?
4. How can we confirm that our factorizations are correct?
5. What is the role of the discriminant in determining the factors of a quadratic?

Tip

When factoring quadratics, always double-check by expanding the factors to ensure they match the original expression.