To simplify the given rational expression:
[tex]\[
\frac{x^2 + 4xy - 5y^2}{3x^2 + 14xy - 5y^2}
\][/tex]
we need to factor both the numerator and the denominator. Let's proceed step-by-step:
### Step 1: Factor the Numerator
The numerator is:
[tex]\[
x^2 + 4xy - 5y^2
\][/tex]
We look for two binomials that multiply to give this quadratic expression. By inspection, we can factorize it as:
[tex]\[
x^2 + 4xy - 5y^2 = (x + 5y)(x - y)
\][/tex]
### Step 2: Factor the Denominator
The denominator is:
[tex]\[
3x^2 + 14xy - 5y^2
\][/tex]
Similarly, we need to factorize this quadratic expression into two binomials. By inspection, we get:
[tex]\[
3x^2 + 14xy - 5y^2 = (3x - y)(x + 5y)
\][/tex]
### Step 3: Combine and Simplify the Factored Expression
After factoring, our expression becomes:
[tex]\[
\frac{(x + 5y)(x - y)}{(3x - y)(x + 5y)}
\][/tex]
We notice that the term [tex]\((x + 5y)\)[/tex] appears in both the numerator and the denominator. Therefore, we can cancel out this common term:
[tex]\[
\frac{(x + 5y)(x - y)}{(3x - y)(x + 5y)} = \frac{x - y}{3x - y}
\][/tex]
Thus, the simplified form of the given rational expression is:
[tex]\[
\frac{x - y}{3x - y}
\][/tex]
### Conclusion
The simplified expression is:
[tex]\[
\boxed{\frac{x - y}{3x - y}}
\][/tex]
So the correct choice is:
A. [tex]\(\frac{x^2+4 x y-5 y^2}{3 x^2+14 x y-5 y^2} = \boxed{\frac{x - y}{3x - y}}\)[/tex]
