A rectangular plot of land has an area of 180 square meters. The length of the plot is 3 meters more than its width. Find the dimensions of the plot.

Let the width of the rectangular plot be $x$ meters.
The length of the plot is 3 meters more than its width, so the length is $(x+3)$ meters.
The area of a rectangle is given by the formula: Area = Length $\times$ Width.
Given that the area is $180 \text{ m}^2$:

$$x(x+3) = 180$$

Expand the equation:

$$x^2 + 3x = 180$$

Rearrange into the standard quadratic form $ax^2 + bx + c = 0$:

$$x^2 + 3x - 180 = 0$$

This is the quadratic equation representing the situation.

We need to find two numbers that multiply to $-180$ and add up to 3. These numbers are 15 and $-12$.
Rewrite the middle term using these numbers:

$$x^2 + 15x - 12x - 180 = 0$$

Factor by grouping:

$$x(x+15) - 12(x+15) = 0$$ $$(x-12)(x+15) = 0$$

Set each factor to zero to find the possible values for $x$:

$$x-12 = 0 \quad \text{or} \quad x+15 = 0$$ $$x = 12 \quad \text{or} \quad x = -15$$

Since $x$ represents the width of the plot, it must be a positive value. Therefore, $x = 12$ meters.

Width $= x = 12$ meters.
Length $= x+3 = 12+3 = 15$ meters.

The dimensions of the plot are:
Width: $\boxed{\text{12 m}}$
Length: $\boxed{\text{15 m}}$