Question

e) ( m(m + 2) + 3(m^2 - 2m - 1) - (m + 2)(m - 2) )

1. use a polynomial to express the area of the following figures.

a) rectangle with length ( 3x + 5 ), width ( 2x - 5 )
b) triangle with base ( 2x + 4 ), height ( 3x - 1 )

Explanation:

Part (a) - Area of the Rectangle

To find the area of a rectangle, we use the formula \( \text{Area} = \text{length} \times \text{width} \). Here, the length is \( 3x + 5 \) and the width is \( 2x - 5 \).

Step 1: Apply the rectangle area formula

The area \( A \) of the rectangle is given by the product of its length and width:
\[
A = (3x + 5)(2x - 5)
\]

Step 2: Expand the product using the distributive property (FOIL method)

First, multiply the First terms: \( 3x \times 2x = 6x^2 \)
Then, the Outer terms: \( 3x \times (-5) = -15x \)
Next, the Inner terms: \( 5 \times 2x = 10x \)
Finally, the Last terms: \( 5 \times (-5) = -25 \)

Combine these terms:
\[
A = 6x^2 - 15x + 10x - 25
\]

Step 3: Combine like terms

Combine the \( x \)-terms: \( -15x + 10x = -5x \)
\[
A = 6x^2 - 5x - 25
\]

Part (b) - Area of the Triangle

To find the area of a triangle, we use the formula \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). Here, the base is \( 2x + 4 \) and the height is \( 3x - 1 \).

Step 1: Apply the triangle area formula

The area \( A \) of the triangle is given by:
\[
A = \frac{1}{2} \times (2x + 4) \times (3x - 1)
\]

Step 2: First, multiply the two binomials \( (2x + 4)(3x - 1) \)

Using the distributive property (FOIL method):
First terms: \( 2x \times 3x = 6x^2 \)
Outer terms: \( 2x \times (-1) = -2x \)
Inner terms: \( 4 \times 3x = 12x \)
Last terms: \( 4 \times (-1) = -4 \)

Combine these terms:
\[
(2x + 4)(3x - 1) = 6x^2 - 2x + 12x - 4
\]

Step 3: Combine like terms in the product

Combine the \( x \)-terms: \( -2x + 12x = 10x \)
\[
(2x + 4)(3x - 1) = 6x^2 + 10x - 4
\]

Step 4: Multiply by \( \frac{1}{2} \)

\[
A = \frac{1}{2}(6x^2 + 10x - 4)
\]
Distribute the \( \frac{1}{2} \) to each term:
\[
A = \frac{1}{2} \times 6x^2 + \frac{1}{2} \times 10x - \frac{1}{2} \times 4
\]
Simplify each term:
\[
A = 3x^2 + 5x - 2
\]

Final Answers

(a) The area of the rectangle is \( \boldsymbol{6x^2 - 5x - 25} \).
(b) The area of the triangle is \( \boldsymbol{3x^2 + 5x - 2} \).

Answer:

Part (a) - Area of the Rectangle

To find the area of a rectangle, we use the formula \( \text{Area} = \text{length} \times \text{width} \). Here, the length is \( 3x + 5 \) and the width is \( 2x - 5 \).

Step 1: Apply the rectangle area formula

The area \( A \) of the rectangle is given by the product of its length and width:
\[
A = (3x + 5)(2x - 5)
\]

Step 2: Expand the product using the distributive property (FOIL method)

First, multiply the First terms: \( 3x \times 2x = 6x^2 \)
Then, the Outer terms: \( 3x \times (-5) = -15x \)
Next, the Inner terms: \( 5 \times 2x = 10x \)
Finally, the Last terms: \( 5 \times (-5) = -25 \)

Combine these terms:
\[
A = 6x^2 - 15x + 10x - 25
\]

Step 3: Combine like terms

Combine the \( x \)-terms: \( -15x + 10x = -5x \)
\[
A = 6x^2 - 5x - 25
\]

Part (b) - Area of the Triangle

To find the area of a triangle, we use the formula \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). Here, the base is \( 2x + 4 \) and the height is \( 3x - 1 \).

Step 1: Apply the triangle area formula

The area \( A \) of the triangle is given by:
\[
A = \frac{1}{2} \times (2x + 4) \times (3x - 1)
\]

Step 2: First, multiply the two binomials \( (2x + 4)(3x - 1) \)

Using the distributive property (FOIL method):
First terms: \( 2x \times 3x = 6x^2 \)
Outer terms: \( 2x \times (-1) = -2x \)
Inner terms: \( 4 \times 3x = 12x \)
Last terms: \( 4 \times (-1) = -4 \)

Combine these terms:
\[
(2x + 4)(3x - 1) = 6x^2 - 2x + 12x - 4
\]

Step 3: Combine like terms in the product

Combine the \( x \)-terms: \( -2x + 12x = 10x \)
\[
(2x + 4)(3x - 1) = 6x^2 + 10x - 4
\]

Step 4: Multiply by \( \frac{1}{2} \)

\[
A = \frac{1}{2}(6x^2 + 10x - 4)
\]
Distribute the \( \frac{1}{2} \) to each term:
\[
A = \frac{1}{2} \times 6x^2 + \frac{1}{2} \times 10x - \frac{1}{2} \times 4
\]
Simplify each term:
\[
A = 3x^2 + 5x - 2
\]

Final Answers

(a) The area of the rectangle is \( \boldsymbol{6x^2 - 5x - 25} \).
(b) The area of the triangle is \( \boldsymbol{3x^2 + 5x - 2} \).