19. $$ (4 m-n)\left(2 m^{2}-3 m n-n^{2}\right) $$ 11. 13. $$ (2 x-y)(2 x+y)(3 x-5 x y-y) $$ $$ 15-n)\left(3 m^{2}-m n+2 n^{2}\right) $$

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UpStudy AI · Accepted Answer

Sure, let's factor each of the given expressions fully.

---

### **Problem 19**

**Given Expression:**
$$
(4m - n)\left(2m^{2} - 3mn - n^{2}\right)
$$

**Factoring the Quadratic Portion:**
We need to factor the quadratic $$2m^{2} - 3mn - n^{2}$$.

1. **Find two numbers** that multiply to $$2m^{2} \times (-n^{2}) = -2m^{2}n^{2}$$ and add up to $$-3mn$$.

2. After testing possible factor pairs, we find that $$(2m + n)$$ and $$(m - n)$$ work:
   $$
   2m^{2} - 3mn - n^{2} = (2m + n)(m - n)
   $$

**Fully Factored Form:**
$$
(4m - n)(2m + n)(m - n)
$$

---

### **Problem 13**

**Given Expression:**
$$
(2x - y)(2x + y)(3x - 5xy - y)
$$

**Analyzing the Third Factor:**
Let's examine the third factor $$3x - 5xy - y$$.

1. **Factor out** common terms:
   $$
   3x - 5xy - y = x(3 - 5y) - y
   $$
   
2. **Check for further factoring:** The expression $$x(3 - 5y) - y$$ does not factor neatly with integer coefficients, so it's already in its simplest form.

**Fully Factored Form:**
$$
(2x - y)(2x + y)(3x - 5xy - y)
$$
*No further factoring is possible for this expression.*

---

### **Problem 15**

**Given Expression:**
$$
-n\left(3m^{2} - mn + 2n^{2}\right)
$$

**Factoring the Quadratic Portion:**
We need to factor $$3m^{2} - mn + 2n^{2}$$.

1. **Calculate the discriminant** to check if it can be factored:
   $$
   \Delta = b^{2} - 4ac = (-n)^{2} - 4(3)(2n^{2}) = n^{2} - 24n^{2} = -23n^{2}
   $$
   
2. **Since the discriminant is negative ($$\Delta = -23n^{2}$$),** the quadratic does not factor over the real numbers.

**Fully Factored Form:**
$$
-n(3m^{2} - mn + 2n^{2})
$$
*No further factoring is possible over the real numbers.*

---

If you have **Problem 11** or any other questions you'd like assistance with, feel free to provide the details!
Sure, here are the simplified answers for each problem: --- **Problem 19:** $$ (4m - n)(2m + n)(m - n) $$ **Problem 13:** $$ (2x - y)(2x + y)(3x - 5xy - y) $$ *Cannot be factored further.* **Problem 15:** $$ -n(3m^{2} - mn + 2n^{2}) $$ *Cannot be factored further over the real numbers.* If you have **Problem 11** or any other questions, feel free to ask!

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