The area and width of a rectangle is $(19 - 3∕5) \text{ cm}^2$ and $(2 + 2∕5) \text{ cm}$ respectively.
Express the length of the rectangle in the form of $(a∕5 + b) \text{ cm}$ where $a$ and $b$ are rational numbers.

1. Set up the expression for the length of the rectangle

$\text{Length} = \frac{\text{Area}}{\text{Width}} = \frac{19 - 3\sqrt{5}}{2 + 2\sqrt{5}}$

2. Multiply the numerator and denominator by the conjugate of the denominator

$\text{Length} = \frac{19 - 3\sqrt{5}}{2 + 2\sqrt{5}} \times \frac{2 - 2\sqrt{5}}{2 - 2\sqrt{5}}$

3. Expand the numerator and denominator

$\text{Numerator} = (19)(2) + (19)(-2\sqrt{5}) + (-3\sqrt{5})(2) + (-3\sqrt{5})(-2\sqrt{5}) = 38 - 38\sqrt{5} - 6\sqrt{5} + 30 = 68 - 44\sqrt{5}$
$\text{Denominator} = 2^2 - (2\sqrt{5})^2 = 4 - 20 = -16$

4. Simplify the fraction to the required form

$\text{Length} = \frac{68 - 44\sqrt{5}}{-16} = \frac{68}{-16} - \frac{44\sqrt{5}}{-16} = -\frac{17}{4} + \frac{11}{4}\sqrt{5}$