Sure! Let's explore the algebraic expressions provided and demonstrate their equivalence. In this case, we're given two expressions that represent the area of Bella's family's patio. We'll break down the steps to show that both expressions indeed represent the same area.
### Step-by-Step Solution:
#### Given:
1. The width of the patio is [tex]\( w \)[/tex].
2. The length of the patio is [tex]\( w + 5 \)[/tex].
#### Area Calculation:
The area of a rectangle is given by multiplying its width by its length.
1. Expression 1:
- One way to write down the area is directly as:
[tex]\[
\text{Area} = w \times (w + 5)
\][/tex]
- This expression states that the area is the product of the width and the length:
[tex]\[
w(w + 5)
\][/tex]
2. Expression 2:
- Another way to represent the area is given:
[tex]\[
\text{Area} = w^2 + 5w
\][/tex]
- This represents the area as a polynomial expression.
#### Simplification to Show Equivalence:
Now, we'll simplify [tex]\( w(w + 5) \)[/tex] to verify that it is equivalent to [tex]\( w^2 + 5w \)[/tex].
1. Starting with the product expression:
[tex]\[
w(w + 5)
\][/tex]
2. Distribute the multiplication:
- Distribute [tex]\( w \)[/tex] across the terms inside the parentheses:
[tex]\[
w \cdot w + w \cdot 5
\][/tex]
- This results in:
[tex]\[
w^2 + 5w
\][/tex]
3. Compare with the polynomial expression:
- We see that both [tex]\( w \cdot w + w \cdot 5 \)[/tex] and [tex]\( w^2 + 5w \)[/tex] simplify to the same form:
[tex]\[
w^2 + 5w
\][/tex]
### Conclusion:
Hence, we have demonstrated through simplification that both algebraic expressions:
1. [tex]\( w(w + 5) \)[/tex]
2. [tex]\( w^2 + 5w \)[/tex]
are equivalent, effectively representing the area of the patio.
Thus, both expressions are just different ways of writing the same area. The polynomial [tex]\( w^2 + 5w \)[/tex] is the simplified and expanded form of the product [tex]\( w(w + 5) \)[/tex].
