1. **State the problem:** Simplify or analyze the expression $3\frac{6}{25} x^2 - \frac{9}{16} y^2$. 2. **Convert mixed number to improper fraction:** $$3\frac{6}{25} = \frac{3 \times 25 + 6}{25} = \frac{75 + 6}{25} = \frac{81}{25}$$ So the expression becomes: $$\frac{81}{25} x^2 - \frac{9}{16} y^2$$ 3. **Recognize the expression form:** This is a difference of two squares type expression, since it is of the form $a x^2 - b y^2$. 4. **Factor the expression:** Write each term as a square: $$\frac{81}{25} x^2 = \left(\frac{9}{5} x\right)^2$$ $$\frac{9}{16} y^2 = \left(\frac{3}{4} y\right)^2$$ So the expression is: $$\left(\frac{9}{5} x\right)^2 - \left(\frac{3}{4} y\right)^2$$ Using difference of squares formula $A^2 - B^2 = (A - B)(A + B)$: $$\left(\frac{9}{5} x - \frac{3}{4} y\right) \left(\frac{9}{5} x + \frac{3}{4} y\right)$$ 5. **Final factored form:** $$\boxed{\left(\frac{9}{5} x - \frac{3}{4} y\right) \left(\frac{9}{5} x + \frac{3}{4} y\right)}$$