A living room is $50m^2$.
This is the surface area.
The length of the living room is $10$ meters
longer than its width.
This information relates two unknowns, the length and the width. So only one quantity of the two is really unknown and the other can be reconstructed using this relationship.
$50 = (x + 10) \times\ x$
We see that the sample solution chose to make the width the unknown, calling it $x$, and expressing the length as $x+10$.
Using the relationship that a rectangle has the area length times width, we indeed come up with the above equation.
According to my book this can be simplified which makes sense.
Yes, you can multiply it out.
50 = (x + 10) \times x = x \times x + 10 \times x = x^2 + 10 x
This is a quadratic equation in the unknown $x$, which can be rewritten as
x^2 + 10 x - 50 = 0
The polynomial on the left hand side is of degree $2$.