To find the value of \( x \), we start with the area of the rectangle given as:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Substituting the expressions for length and width:
\[ 20 = (x + 4)(x - 4) \]
We recognize that \( (x + 4)(x - 4) \) is a difference of squares, which simplifies to:
\[ x^2 - 16 \]
Now we can set up the equation:
\[ 20 = x^2 - 16 \]
Next, we will add 16 to both sides:
\[ 20 + 16 = x^2 \]
This simplifies to:
\[ 36 = x^2 \]
To solve for \( x \), we take the square root of both sides:
\[ x = \sqrt{36} \quad \text{or} \quad x = -\sqrt{36} \] \[ x = 6 \quad \text{or} \quad x = -6 \]
Since \( x \) represents a length, we discard the negative value. Therefore, the value of \( x \) is:
\[ \boxed{6} \text{ cm} \]