Certainly! We need to perform the division of the given expression:
[tex]\[
\frac{-15 x^2 - 8 y^2 + 22 x y}{2 y - 3 x}
\][/tex]
Let's break this down step-by-step.
1. Define the numerator and the denominator:
- The numerator of our fraction is [tex]\(-15 x^2 - 8 y^2 + 22 x y\)[/tex].
- The denominator of our fraction is [tex]\(2 y - 3 x\)[/tex].
2. Express the fraction in terms of the given numerator and denominator:
[tex]\[
\frac{-15 x^2 - 8 y^2 + 22 x y}{2 y - 3 x}
\][/tex]
3. Check for common factors:
- Verify if there is a common factor between the numerator and the denominator which can be factored out and canceled. In this case, there are no obvious common factors between [tex]\(-15 x^2 - 8 y^2 + 22 x y\)[/tex] and [tex]\(2 y - 3 x\)[/tex].
4. Simplify the fraction (if possible):
- Since there are no common factors in the numerator and the denominator that can be factored out, the fraction remains in its original form.
The final simplified form of the given expression is:
[tex]\[
\frac{-15 x^2 - 8 y^2 + 22 x y}{-3 x + 2 y}
\][/tex]
