$$ 912 m ^ { 2 } - 19 m n + 5 n ^ { 2 } $$

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UpStudy AI · Accepted Answer

The expression you've provided is a quadratic in terms of variables $$ m $$ and $$ n $$:

$$
912m^2 - 19mn + 5n^2
$$

### **Objective: Factor the Expression**

To factor this quadratic expression, we look for two binomials of the form:

$$
(am + bn)(cm + dn)
$$

When expanded, this yields:

$$
ac\,m^2 + (ad + bc)\,mn + bd\,n^2
$$

By comparing coefficients, we establish the following system of equations:

1. $$ ac = 912 $$          *(Coefficient of $$ m^2 $$)*
2. $$ ad + bc = -19 $$      *(Coefficient of $$ mn $$)*
3. $$ bd = 5 $$           *(Coefficient of $$ n^2 $$)*

#### **Step 1: Find Possible Pairs for $$ a $$ and $$ c $$**

First, factorize 912:

$$
912 = 2^4 \times 3 \times 19
$$

Possible integer pairs $$(a, c)$$ such that $$ ac = 912 $$:

- $$ (1, 912) $$
- $$ (2, 456) $$
- $$ (3, 304) $$
- $$ (4, 228) $$
- $$ (6, 152) $$
- $$ (8, 114) $$
- $$ (12, 76) $$
- $$ (16, 57) $$
- $$ (19, 48) $$
- $$ (24, 38) $$
- ...and their negative counterparts.

#### **Step 2: Find Possible Pairs for $$ b $$ and $$ d $$**

Since $$ bd = 5 $$ and 5 is a prime number, the possible integer pairs are:

- $$ (1, 5) $$
- $$ (-1, -5) $$

#### **Step 3: Check for Compatibility**

For each combination of $$(a, c)$$ and $$(b, d)$$, check if $$ ad + bc = -19 $$.

**Example Attempt:**

Let's try $$ b = 1 $$ and $$ d = 5 $$:

1. $$ ad + bc = a \times 5 + c \times 1 = 5a + c = -19 $$
2. From $$ ac = 912 $$, suppose $$ a = 19 $$, then $$ c = 48 $$.
3. Check $$ 5a + c = 5 \times 19 + 48 = 95 + 48 = 143 \neq -19 $$

This doesn't satisfy the second equation. Repeating this process for all possible pairs reveals that **no integer solutions satisfy all three equations simultaneously**.

### **Conclusion**

The quadratic expression:

$$
912m^2 - 19mn + 5n^2
$$

**is not factorable** into binomials with integer coefficients. This means the expression is **irreducible** over the set of integers.

### **Alternative Approach: Using the Quadratic Formula**

If you consider the expression as a quadratic in $$ m $$, you can explore its roots using the quadratic formula:

$$
m = \frac{19n \pm \sqrt{(19n)^2 - 4 \times 912 \times 5n^2}}{2 \times 912} = \frac{19n \pm \sqrt{-17879n^2}}{1824}
$$

The discriminant here is negative ($$ -17879n^2 $$), indicating that the quadratic does not have real roots. This further confirms that the expression cannot be factored into real linear factors.

### **Final Answer**

The expression $$ 912m^2 - 19mn + 5n^2 $$ cannot be factored into simpler binomial terms with integer coefficients. It is **irreducible** over the integers.
The expression $$ 912m^2 - 19mn + 5n^2 $$ cannot be factored into simpler binomial terms with integer coefficients. It is **irreducible** over the integers.

\( 912 m ^ { 2 } - 19 m n + 5 n ^ { 2 } \)

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