(x+1)(x−4)(x+5)

(x−3)(x+2)(x−6)(x - 3)(x + 2)(x - 6)(x−3)(x+2)(x−6)

(2x+5)(x−2)(x+3)(2x + 5)(x - 2)(x + 3)(2x+5)(x−2)(x+3)

(x+7)(x−1)(x+2)(x + 7)(x - 1)(x + 2)(x+7)(x−1)(x+2)

(3x+2)(x−5)(x+4)(3x + 2)(x - 5)(x + 4)(3x+2)(x−5)(x+4)

(x−2)(x+3)2(x - 2)(x + 3)^2(x−2)(x+3)2

(2x−1)(x+3)(x−4)(2x - 1)(x + 3)(x - 4)(2x−1)(x+3)(x−4)

(x−7)(x+4)2(x - 7)(x + 4)^2(x−7)(x+4)2

(x+2)(x−3)(x+6)(x + 2)(x - 3)(x + 6)(x+2)(x−3)(x+6)

(4x+1)(x−2)(x+3)(4x + 1)(x - 2)(x + 3)(4x+1)(x−2)(x+3)

You’ve listed several polynomial expressions. I can help you expand or simplify them. Let’s work through each one step by step.

1. **(x + 1)(x − 4)(x + 5):**
   - Expand \( (x + 1)(x − 4) \) first:
     \[
     (x + 1)(x − 4) = x^2 − 4x + x − 4 = x^2 − 3x − 4
     \]
   - Then multiply by \( (x + 5) \):
     \[
     (x^2 − 3x − 4)(x + 5) = x^3 + 5x^2 − 3x^2 − 15x − 4x − 20
     \]
     Simplified: 
     \[
     x^3 + 2x^2 − 19x − 20
     \]

2. **(x − 3)(x + 2)(x − 6):**
   - Expand \( (x − 3)(x + 2) \):
     \[
     (x − 3)(x + 2) = x^2 + 2x − 3x − 6 = x^2 − x − 6
     \]
   - Then multiply by \( (x − 6) \):
     \[
     (x^2 − x − 6)(x − 6) = x^3 − 6x^2 − x^2 + 6x − 6x − 36 
     \]
     Simplified:
     \[
     x^3 − 7x^2 + 6x − 36
     \]

3. **(2x + 5)(x − 2)(x + 3):**
   - Expand \( (x − 2)(x + 3) \):
     \[
     (x − 2)(x + 3) = x^2 + 3x − 2x − 6 = x^2 + x − 6
     \]
   - Then multiply by \( (2x + 5) \):
     \[
     (2x + 5)(x^2 + x − 6) = 2x^3 + 2x^2 − 12x + 5x^2 + 5x − 30 
     \]
     Simplified:
     \[
     2x^3 + 7x^2 − 7x − 30
     \]

4. **(x + 7)(x − 1)(x + 2):**
   - Expand \( (x + 7)(x − 1) \):
     \[
     (x + 7)(x − 1) = x^2 − x + 7x − 7 = x^2 + 6x − 7
     \]
   - Then multiply by \( (x + 2) \):
     \[
     (x^2 + 6x − 7)(x + 2) = x^3 + 2x^2 + 6x^2 + 12x − 7x − 14
     \]
     Simplified:
     \[
     x^3 + 8x^2 − 5x − 14
     \]

5. **(3x + 2)(x − 5)(x + 4):**
   - Expand \( (x − 5)(x + 4) \):
     \[
     (x − 5)(x + 4) = x^2 + 4x − 5x − 20 = x^2 − x − 20
     \]
   - Then multiply by \( (3x + 2) \):
     \[
     (3x + 2)(x^2 − x − 20) = 3x^3 − 3x^2 − 60x + 2x^2 − 2x − 40
     \]
     Simplified:
     \[
     3x^3 − x^2 − 62x − 40
     \]

6. **(x − 2)(x + 3)^2:**
   - Expand \( (x + 3)^2 \):
     \[
     (x + 3)^2 = x^2 + 6x + 9
     \]
   - Then multiply by \( (x − 2) \):
     \[
     (x − 2)(x^2 + 6x + 9) = x^3 + 6x^2 + 9x − 2x^2 − 12x − 18
     \]
     Simplified:
     \[
     x^3 + 4x^2 − 3x − 18
     \]

7. **(2x − 1)(x + 3)(x − 4):**
   - Expand \( (x + 3)(x − 4) \):
     \[
     (x + 3)(x − 4) = x^2 − 4x + 3x − 12 = x^2 − x − 12
     \]
   - Then multiply by \( (2x − 1) \):
     \[
     (2x − 1)(x^2 − x − 12) = 2x^3 − 2x^2 − 24x − x^2 + x + 12
     \]
     Simplified:
     \[
     2x^3 − 3x^2 − 23x + 12
     \]

8. **(x − 7)(x + 4)^2:**
   - Expand \( (x + 4)^2 \):
     \[
     (x + 4)^2 = x^2 + 8x + 16
     \]
   - Then multiply by \( (x − 7) \):
     \[
     (x − 7)(x^2 + 8x + 16) = x^3 + 8x^2 + 16x − 7x^2 − 56x − 112
     \]
     Simplified:
     \[
     x^3 + x^2 − 40x − 112
     \]

9. **(x + 2)(x − 3)(x + 6):**
   - Expand \( (x + 2)(x − 3) \):
     \[
     (x + 2)(x − 3) = x^2 − 3x + 2x − 6 = x^2 − x − 6
     \]
   - Then multiply by \( (x + 6) \):
     \[
     (x^2 − x − 6)(x + 6) = x^3 + 6x^2 − x^2 − 6x − 6x − 36
     \]
     Simplified:
     \[
     x^3 + 5x^2 − 12x − 36
     \]

10. **(4x + 1)(x − 2)(x + 3):**
    - Expand \( (x − 2)(x + 3) \):
      \[
      (x − 2)(x + 3) = x^2 + 3x − 2x − 6 = x^2 + x − 6
      \]
    - Then multiply by \( (4x + 1) \):
      \[
      (4x + 1)(x^2 + x − 6) = 4x^3 + 4x^2 − 24x + x^2 + x − 6
      \]
      Simplified:
      \[
      4x^3 + 5x^2 − 23x − 6
      \]

Feel free to ask for any more details or clarifications. 

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Here are 5 related questions:
1. What is the factored form of \(x^3 + 2x^2 − 19x − 20\)?
2. Can you find the roots of \(x^3 − 7x^2 + 6x − 36\)?
3. How do you use synthetic division to simplify polynomials?
4. What are the real and complex roots of \(2x^3 − 7x − 30\)?
5. How does the discriminant help in finding roots of quadratic expressions?

**Tip:** When expanding polynomials, keep track of signs and like terms carefully to avoid mistakes.

Explore how to expand and simplify polynomial expressions like (x + 1)(x − 4)(x + 5) and more. Get step-by-step solutions with detailed explanations and examples. Suitable for high school students studying advanced algebra.

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