To find the equations for the perimeter and the area of the rectangle, let's use the given dimensions:

• Length ($L$): $5a + 2b$
• Width ($W$): $4a - 3b$

Perimeter of the Rectangle

The perimeter $P$ of a rectangle is given by: $P = 2L + 2W$ Substituting the values for $L$ and $W$: $P = 2(5a + 2b) + 2(4a - 3b)$ Distribute the 2: $P = 10a + 4b + 8a - 6b$ Now, combine like terms: $P = (10a + 8a) + (4b - 6b) = 18a - 2b$ So, the equation for the perimeter is: $\boxed{P = 18a - 2b}$

Area of the Rectangle

The area $A$ of a rectangle is given by: $A = L \times W$ Substituting the values for $L$ and $W$: $A = (5a + 2b)(4a - 3b)$ Apply the distributive property (FOIL method): $A = (5a)(4a) + (5a)(-3b) + (2b)(4a) + (2b)(-3b)$ Simplify each term: $A = 20a^2 - 15ab + 8ab - 6b^2$ Now, combine like terms: $A = 20a^2 - 7ab - 6b^2$ So, the equation for the area is: $\boxed{A = 20a^2 - 7ab - 6b^2}$

Would you like more details or have any other questions?

Here are some related questions:

1. What is the perimeter when $a = 2$ and $b = 1$?
2. What is the area when $a = 1$ and $b = 2$?
3. How would the perimeter change if the width were $4a + 3b$ instead of $4a - 3b$?
4. What happens to the area if both $a$ and $b$ are doubled?
5. Can you find the length and width if the perimeter is given as $36a + 2b$?

Tip: When multiplying binomials, remember to use the distributive property carefully to ensure all terms are accounted for.