Sure, let's find the simplified form of the expression [tex]\(\frac{9 x^2 - 25 y^2}{6 x^2 + 19 x y + 15 y}\)[/tex].
1. Identify the components:
- Numerator: [tex]\(9 x^2 - 25 y^2\)[/tex]
- Denominator: [tex]\(6 x^2 + 19 x y + 15 y\)[/tex]
2. Factor the numerator and denominator where possible:
- The numerator [tex]\(9 x^2 - 25 y^2\)[/tex] is a difference of squares.
- The difference of squares formula is [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].
So, we can write:
[tex]\[
9 x^2 - 25 y^2 = (3x)^2 - (5y)^2 = (3x - 5y)(3x + 5y)
\][/tex]
- The denominator [tex]\(6x^2 + 19xy + 15y\)[/tex] is a polynomial in two variables.
3. Simplify the expression using the factors:
[tex]\[
\frac{9 x^2 - 25 y^2}{6 x^2 + 19 x y + 15 y} = \frac{(3x - 5y)(3x + 5y)}{6 x^2 + 19 x y + 15 y}
\][/tex]
4. Check for cancellation of common factors:
- Observe both the numerator and the denominator to see if any factors are common and could be canceled out. In this specific case, there are no common factors that can be simplified further in the given denominator and numerator.
Thus, the expression simplifies to:
[tex]\[
\boxed{\frac{9 x^2 - 25 y^2}{6 x^2 + 19 x y + 15 y}}
\][/tex]
