$$ 6 x ^ { 2 } + 5 x y - 15 y ^ { 2 } $$

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The expression you've provided is a quadratic in two variables:

$$
6x^2 + 5xy - 15y^2
$$

### **Attempting to Factor the Expression**

To factor a quadratic trinomial of the form $$ ax^2 + bxy + cy^2 $$, we look for two binomials:

$$
(ax + by)(cx + dy)
$$

Expanding these binomials gives:

$$
acx^2 + (ad + bc)xy + bdy^2
$$

For your expression:
- $$ ac = 6 $$
- $$ bd = -15 $$
- $$ ad + bc = 5 $$

**Possible Factor Pairs:**
1. **For $$ ac = 6 $$:**
   - (2, 3)
   - (3, 2)
   - (6, 1)
   - (1, 6)

2. **For $$ bd = -15 $$:**
   - (1, -15)
   - (-1, 15)
   - (3, -5)
   - (-3, 5)
   - (5, -3)
   - (-5, 3)

**Testing Factor Combinations:**

After systematically testing all possible combinations of these factors, none satisfy the condition $$ ad + bc = 5 $$. For example:

- **Using $$ a = 2 $$ and $$ c = 3 $$:**
  - If $$ b = 5 $$ and $$ d = -3 $$:
    $$
    ad + bc = 2(-3) + 3(5) = -6 + 15 = 9 \neq 5
    $$
  - Similarly, other combinations do not yield $$ ad + bc = 5 $$.

**Conclusion:**

Since none of the integer or simple fractional combinations satisfy all the necessary conditions for factoring, the quadratic expression:

$$
6x^2 + 5xy - 15y^2
$$

**cannot be factored into binomials with rational coefficients**.

### **Alternative Approach: Completing the Square**

If you're interested in other forms, such as completing the square or analyzing the expression further, please let me know!
The expression $$6x^2 + 5xy - 15y^2$$ cannot be factored into binomials with rational coefficients.

\( 6 x ^ { 2 } + 5 x y - 15 y ^ { 2 } \)

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