1. **State the problem:** We are given a rectangle with width $5a^{2}b^{4}$ and height $3ab^{2}$. We need to find the perimeter $P$ and area $A$ of this rectangle. 2. **Formulas:** - Perimeter of a rectangle: $$P = 2(\text{width} + \text{height})$$ - Area of a rectangle: $$A = \text{width} \times \text{height}$$ 3. **Calculate the perimeter:** $$P = 2(5a^{2}b^{4} + 3ab^{2})$$ 4. **Simplify inside the parentheses:** $$5a^{2}b^{4} + 3ab^{2}$$ 5. **Factor out the common term $ab^{2}$:** $$5a^{2}b^{4} + 3ab^{2} = ab^{2}(5ab^{2} + 3)$$ 6. **Substitute back into perimeter formula:** $$P = 2 \times ab^{2}(5ab^{2} + 3)$$ 7. **Final expression for perimeter:** $$P = 2ab^{2}(5ab^{2} + 3)$$ 8. **Calculate the area:** $$A = (5a^{2}b^{4})(3ab^{2})$$ 9. **Multiply coefficients and variables:** $$A = 5 \times 3 \times a^{2} \times a \times b^{4} \times b^{2} = 15a^{3}b^{6}$$ 10. **Final answers:** - Perimeter: $$P = 2ab^{2}(5ab^{2} + 3)$$ - Area: $$A = 15a^{3}b^{6}$$