Let's carefully analyze the problem and the expressions to determine which ones are equivalent to the given perimeter expression for the larger garden.
### Step-by-step analysis of the given expression:
The expression given represents the perimeter of the larger garden, where the length and width are each increased by [tex]\( x \)[/tex] feet:
[tex]\[
(x + 15) + (x + 5) + (x + 15) + (x + 5)
\][/tex]
First, we combine like terms:
[tex]\[
(x + 15) + (x + 5) + (x + 15) + (x + 5) = 4x + 40
\][/tex]
This simplifies the given perimeter expression to [tex]\( 4x + 40 \)[/tex].
### Checking the equivalency of each given expression:
#### Expression A: [tex]\(4x + 40\)[/tex]
This is exactly what we derived after simplifying the initial expression. Therefore, Expression A is equivalent.
#### Expression B: [tex]\(2(2x + 20)\)[/tex]
Let's simplify this expression:
[tex]\[
2(2x + 20) = 2 \cdot 2x + 2 \cdot 20 = 4x + 40
\][/tex]
This matches our simplified form [tex]\( 4x + 40 \)[/tex]. Therefore, Expression B is equivalent.
#### Expression C: [tex]\(2(x + 15)(x + 5)\)[/tex]
Let's analyze this expression:
[tex]\[
2(x + 15)(x + 5)
\][/tex]
This is a quadratic expression and represents an area calculation rather than a perimeter. Upon expansion:
[tex]\[
2(x + 15)(x + 5) = 2(x^2 + 20x + 75) = 2x^2 + 40x + 150
\][/tex]
This does not match our simplified perimeter expression [tex]\( 4x + 40 \)[/tex]. Therefore, Expression C is not equivalent.
#### Expression D: [tex]\(4(x + 15)(x + 5)\)[/tex]
This is also a quadratic expression:
[tex]\[
4(x + 15)(x + 5)
\][/tex]
Expanding this:
[tex]\[
4(x + 15)(x + 5) = 4(x^2 + 20x + 75) = 4x^2 + 80x + 300
\][/tex]
This does not match our simplified perimeter expression [tex]\( 4x + 40 \)[/tex]. Therefore, Expression D is not equivalent.
#### Expression E: [tex]\(2(x + 15) + 2(x + 5)\)[/tex]
Let's simplify this expression:
[tex]\[
2(x + 15) + 2(x + 5) = 2x + 30 + 2x + 10 = 4x + 40
\][/tex]
This matches our simplified form [tex]\( 4x + 40 \)[/tex]. Therefore, Expression E is equivalent.
### Conclusion:
The expressions equivalent to the given perimeter expression [tex]\( (x + 15) + (x + 5) + (x + 15) + (x + 5) \)[/tex] are:
- [tex]\(4x + 40\)[/tex] (Expression A)
- [tex]\(2(2x + 20)\)[/tex] (Expression B)
- [tex]\(2(x + 15) + 2(x + 5)\)[/tex] (Expression E)
