Solution:

Let us solve the problem step by step.

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Part (a): Expression for the area of the path in terms of $x$

1. The garden has dimensions $x$ (length) and $x - 2$ (width).
   Its area is:
   $A_{\text{garden}} = x(x - 2) = x^2 - 2x$

2. If a uniform path is built around the garden with a width $w$, the total dimensions (including the path) become:
   $\text{Length: } x + 2w, \quad \text{Width: } (x - 2) + 2w = x - 2 + 2w$

3. The total area including the path is:
   $A_{\text{total}} = (x + 2w)(x - 2 + 2w)$

4. The area of the path is the difference between the total area and the garden's area:
   $A_{\text{path}} = A_{\text{total}} - A_{\text{garden}}$

Expanding $A_{\text{total}}$:
$A_{\text{total}} = (x + 2w)(x - 2 + 2w) = x^2 + 2wx - 2x + 2wx - 4w + 4w^2$ $A_{\text{total}} = x^2 + 4wx - 2x - 4w + 4w^2$

Subtracting $A_{\text{garden}} = x^2 - 2x$:
$A_{\text{path}} = \big(x^2 + 4wx - 2x - 4w + 4w^2\big) - \big(x^2 - 2x\big)$ $A_{\text{path}} = 4wx - 4w + 4w^2$

Thus, the area of the path in terms of $x$ and $w$ is:
$A_{\text{path}} = 4w(x - 1) + 4w^2$

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Part (b): Find the dimensions and area of the garden

We are told that the area of the path is $40 \, \text{m}^2$ and the width of the path is $2 \, \text{m}$:
Substitute $w = 2$ into the expression for $A_{\text{path}}$:
$A_{\text{path}} = 4(2)(x - 1) + 4(2^2)$ $40 = 8(x - 1) + 16$ $40 = 8x - 8 + 16$ $40 = 8x + 8$ $8x = 32$ $x = 4$

1. Dimensions of the garden:
   Length = $x = 4 \, \text{m}$, Width = $x - 2 = 4 - 2 = 2 \, \text{m}$.

2. Area of the garden:
   $A_{\text{garden}} = x(x - 2) = 4(2) = 8 \, \text{m}^2.$

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Part (c): If the width of the path is $2 \, \text{m}$, find the area of the path.

We already solved this in part (b):
$A_{\text{path}} = 40 \, \text{m}^2$

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Part (d): If the garden measures $12 \, \text{m} \times 10 \, \text{m}$, find the area of the path.

1. The area of the garden is:
   $A_{\text{garden}} = 12 \times 10 = 120 \, \text{m}^2$

2. Total dimensions including the path:
   Length = $12 + 2w$, Width = $10 + 2w$.

   Total area including the path:
   $A_{\text{total}} = (12 + 2w)(10 + 2w)$

3. Area of the path is the difference:
   $A_{\text{path}} = A_{\text{total}} - A_{\text{garden}}$ Substitute $w = 2$:
   $A_{\text{total}} = (12 + 4)(10 + 4) = 16 \times 14 = 224 \, \text{m}^2$ $A_{\text{path}} = 224 - 120 = 104 \, \text{m}^2$

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Final Answers:

(a) Expression for the path's area:
$A_{\text{path}} = 4w(x - 1) + 4w^2$

(b) Dimensions of the garden: $4 \, \text{m} \times 2 \, \text{m}$, Area = $8 \, \text{m}^2$.

(c) Area of the path: $40 \, \text{m}^2$.

(d) Area of the path for $12 \, \text{m} \times 10 \, \text{m}$: $104 \, \text{m}^2$.

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Follow-up Questions:

1. How would the expression for the area of the path change for a different width $w$?
2. What happens to the area of the path if $w = 3 \, \text{m}$?
3. Can you verify the total area for any $x$ and $w$ values using the formula?
4. How does the area of the path change as the width $w$ increases?
5. Why does the total area formula include quadratic terms?

Tip:

Always verify your dimensions carefully when dealing with paths around shapes to avoid errors in perimeter or area calculations.