11. $$ p^{2}-72+p $$ 13. $$ x^{2}-x-20 $$ 15. $$ -x^{2}-2 x+63 $$ Compiled by: Motsoeneng L.S

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Certainly! Let's factor each of the provided polynomial expressions step by step.

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### **11. $$ p^{2} + p - 72 $$**

**Objective:** Factor the quadratic expression $$ p^{2} + p - 72 $$.

**Steps:**
1. **Identify the coefficients:**  
   - $$ a = 1 $$ (coefficient of $$ p^2 $$)  
   - $$ b = 1 $$ (coefficient of $$ p $$)  
   - $$ c = -72 $$

2. **Find two numbers that multiply to $$ a \times c = -72 $$** and **add to $$ b = 1 $$.**  
   - The numbers **9** and **-8** satisfy this:  
     $$ 9 \times (-8) = -72 $$  
     $$ 9 + (-8) = 1 $$

3. **Express the middle term using these numbers:**  
   $$ p^{2} + 9p - 8p - 72 $$

4. **Factor by grouping:**  
   - Group the first two and the last two terms:  
     $$ (p^{2} + 9p) + (-8p - 72) $$  
   - Factor out the common terms:  
     $$ p(p + 9) - 8(p + 9) $$  
   - Factor out $$ (p + 9) $$:  
     $$ (p + 9)(p - 8) $$

**Factored Form:**  
$$ p^{2} + p - 72 = (p + 9)(p - 8) $$

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### **13. $$ x^{2} - x - 20 $$**

**Objective:** Factor the quadratic expression $$ x^{2} - x - 20 $$.

**Steps:**
1. **Identify the coefficients:**  
   - $$ a = 1 $$  
   - $$ b = -1 $$  
   - $$ c = -20 $$

2. **Find two numbers that multiply to $$ a \times c = -20 $$** and **add to $$ b = -1 $$.**  
   - The numbers **4** and **-5** satisfy this:  
     $$ 4 \times (-5) = -20 $$  
     $$ 4 + (-5) = -1 $$

3. **Express the middle term using these numbers:**  
   $$ x^{2} + 4x - 5x - 20 $$

4. **Factor by grouping:**  
   - Group the first two and the last two terms:  
     $$ (x^{2} + 4x) + (-5x - 20) $$  
   - Factor out the common terms:  
     $$ x(x + 4) - 5(x + 4) $$  
   - Factor out $$ (x + 4) $$:  
     $$ (x + 4)(x - 5) $$

**Factored Form:**  
$$ x^{2} - x - 20 = (x + 4)(x - 5) $$

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### **15. $$ -x^{2} - 2x + 63 $$**

**Objective:** Factor the quadratic expression $$ -x^{2} - 2x + 63 $$.

**Steps:**
1. **Factor out the negative sign:**  
   $$ - (x^{2} + 2x - 63) $$

2. **Now, factor the quadratic $$ x^{2} + 2x - 63 $$.**

3. **Identify the coefficients:**  
   - $$ a = 1 $$  
   - $$ b = 2 $$  
   - $$ c = -63 $$

4. **Find two numbers that multiply to $$ a \times c = -63 $$** and **add to $$ b = 2 $$.**  
   - The numbers **9** and **-7** satisfy this:  
     $$ 9 \times (-7) = -63 $$  
     $$ 9 + (-7) = 2 $$

5. **Express the middle term using these numbers:**  
   $$ x^{2} + 9x - 7x - 63 $$

6. **Factor by grouping:**  
   - Group the first two and the last two terms:  
     $$ (x^{2} + 9x) + (-7x - 63) $$  
   - Factor out the common terms:  
     $$ x(x + 9) - 7(x + 9) $$  
   - Factor out $$ (x + 9) $$:  
     $$ (x + 9)(x - 7) $$

7. **Don't forget the negative sign factored out initially:**  
   $$ - (x + 9)(x - 7) $$

**Factored Form:**  
$$ -x^{2} - 2x + 63 = -(x + 9)(x - 7) $$

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**Compiled Factored Forms:**

1. **Problem 11:**  
   $$ p^{2} + p - 72 = (p + 9)(p - 8) $$

2. **Problem 13:**  
   $$ x^{2} - x - 20 = (x + 4)(x - 5) $$

3. **Problem 15:**  
   $$ -x^{2} - 2x + 63 = -(x + 9)(x - 7) $$

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I hope this helps! If you have any more questions or need further clarification, feel free to ask.
1. $$ p^{2} + p - 72 = (p + 9)(p - 8) $$ 2. $$ x^{2} - x - 20 = (x + 4)(x - 5) $$ 3. $$ -x^{2} - 2x + 63 = -(x + 9)(x - 7) $$

11. \( p^{2}-72+p \) 13. \( x^{2}-x-20 \) 15. \( -x^{2}-2 x+63 \) Compiled by: Motsoeneng L.S

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