The dancing space in a room has dimensions (x-3) m by (x+3) m. There is a carpeted margin of (2)/(5)x m wide all around. If the total area of the room is 315 m2, calculate the value of x.

The dancing space has dimensions $(x-3)$ m by $(x+3)$ m.
The carpeted margin is $\frac{2}{5}x$ wide all around.
The total length of the room is the length of the dancing space plus twice the margin width:

The total width of the room is the width of the dancing space plus twice the margin width:

The total area of the entire room is $315 \text{ m}^2$.

Multiply both sides by 25:

Add 225 to both sides:

Divide by 81:

Take the square root:

Since $x$ represents a dimension, it must be positive.

The value of $x$ is $\boxed{10}$.

Using $x=10$:
Length of dancing space $L_{\text{dancing}} = x+3 = 10+3 = 13 \text{ m}$
Width of dancing space $W_{\text{dancing}} = x-3 = 10-3 = 7 \text{ m}$
Area of dancing space $A_{\text{dancing}} = L_{\text{dancing}} \times W_{\text{dancing}} = 13 \text{ m} \times 7 \text{ m} = 91 \text{ m}^2$.

The area of the carpeted margin is the total area minus the area of the dancing space.

The area of the carpeted margin is $\boxed{224 \text{ m}^2}$.