Answer: Yes, it is a quadratic equation
We can distribute the outer x to each term inside
x(x-6) = x^2-6x
So the original equation turns into
x^2-6x = 20
Then you could subtract 20 from both sides to get everything to the same side
x^2-6x-20 = 20-20
x^2-6x-20 = 0
This quadratic equation is in the form ax^2+bx+c = 0 with a = 1, b = -6, c = -20.
The presence of the x^2 term, and having it be the largest exponent, is what makes this a quadratic.