So, to check if an equation is a quadratic equation,
you want to make two passes through it (both sides):

Does it have an $\,x^2\,$ term appearing somewhere?
If not, then it's not a quadratic equation.
Note: it can have lots of $\,x^2\,$ terms!

The only other two term types that are allowed are $\,x\,$ terms and constants terms.
(For example: no $\,x^3\,$ terms, no variables inside square roots, no variables in denominators, and so on.)
So, sweep across the equation and look for anything other than $\,x\,$ terms and constant terms.
If you find any, then it's not a quadratic equation.
EXAMPLES:
In this exercise, you will practice identifying quadratic equations.
Question:
Is
$\,x^2 = x + 4\,$ a quadratic equation?
Solution:
Does it have an $\,x^2\,$ term?
Check!
Anything other than $\,x\,$ terms or constant terms?
Nope.
Check!
YES, it is a quadratic equation.
Question:
Is
$\,3x  4 = x + 1\,$ a quadratic equation?
Solution:
Does it have an $\,x^2\,$ term?
Nope.
So, it's not a quadratic equation.
Question:
Is
$\,x  2x^2 = 1 + x^5\,$ a quadratic equation?
Solution:
Does it have an $\,x^2\,$ term?
Check!
Anything other than $\,x\,$ terms or constant terms?
Oops.
Quadratic equations are not allowed to have an $\,x^5\,$ term.
So, it's not a quadratic equation.