Question

1. write and simplify a polynomial expression to represent the area of the rectangle.

3x - 4
x + 5

1. write and simplify a polynomial expression to represent the area of the rectangle.

4a + 5
2 - a

Explanation:

Step1: Recall area formula

The area $A$ of a rectangle is $A = \text{length}\times\text{width}$.

Step2: Calculate area for first rectangle

For the rectangle with length $3x - 4$ and width $5 + x$, we use the FOIL method: $(3x - 4)(x + 5)=3x\times x+3x\times5-4\times x - 4\times5=3x^{2}+15x-4x - 20$.

Step3: Simplify first - rectangle area

Combining like terms, we get $3x^{2}+(15x - 4x)-20=3x^{2}+11x - 20$.

Step4: Calculate area for second rectangle

For the rectangle with length $4a + 5$ and width $2 - a$, we use the FOIL method: $(4a + 5)(2 - a)=4a\times2-4a\times a+5\times2-5\times a=8a-4a^{2}+10 - 5a$.

Step5: Simplify second - rectangle area

Combining like terms, we have $-4a^{2}+(8a - 5a)+10=-4a^{2}+3a + 10$.

Answer:

For the first rectangle, the area is $3x^{2}+11x - 20$.
For the second rectangle, the area is $-4a^{2}+3a + 10$.