1. **State the problem:** Simplify the expression $$-10xy \times -4x^{2}y + \frac{22x^{4}y^{3}}{11xy^{2}}$$. 2. **Apply multiplication and division rules:** - Multiply the first two terms: $$-10xy \times -4x^{2}y = ( -10 \times -4 )(x \times x^{2})(y \times y) = 40x^{3}y^{2}$$. - Simplify the division: $$\frac{22x^{4}y^{3}}{11xy^{2}}$$. 3. **Simplify the division by canceling common factors:** $$\frac{22x^{4}y^{3}}{11xy^{2}} = \frac{\cancel{22}x^{4}y^{3}}{\cancel{11}xy^{2}} = 2 \times \frac{x^{4}}{x} \times \frac{y^{3}}{y^{2}} = 2x^{3}y$$. 4. **Combine the results:** $$40x^{3}y^{2} + 2x^{3}y$$. 5. **Factor common terms:** $$x^{3}y(40y + 2) = 2x^{3}y(20y + 1)$$. **Final simplified expression:** $$2x^{3}y(20y + 1)$$.