How do you determine if `(5y^2)/x^2+4x` is a polynomial and if so, how do you identify if it is a monomial, binomial, or trinomial?

Polynomials are algebraic expressions made up of more than one "term."

A "term" is a cluster of numbers and letters that are connected to each other by multiplication and division.

These are all "terms"
`x`

`3y`

`7m xx 22`

`(4x - 2)^2 -: 3(6x - 13)^6`
.........................

Terms are separated from each other by addition or subtraction.
Here are expressions with two terms:
a + 3

`45x^2 - m^3`

`(4x - 2)^2 -: 3(6x - 13)^6 + (a + x) (a - x)`

`larr`------ one term ----- `rarr` + `larr`another`rarr`
.........................

You can tell if an expression is a monomial, a binomial, a trinomial, or a polynomial by counting the terms.

We give expressions with 1, 2, or 3 terms special names, but after that we stop counting and just say "polynomial."

Specific kinds of polynomials are called:

a monomial if it has exactly 1 term
`3s^3`

a binomial if it has exactly 2 terms
`2x + 3`

a trinomial if it has exactly 3 terms
`x^2 + 6x + 9`
1 . . . . .2 . . . 3 `larr` (the number of terms is 3)

a polynomial if it has more than 3 terms
(But expressions with 1, 2, or 3 terms are also still called polynomials.)

`m` `larr` monomial or polynomial

`x - 3` `larr` binomial or polynomial

`6y^3 - 9y^2 + 3y` `larr` trinomial or polynomial

`8m^3 + 3m^2 - 7m + 3` `larr` polynomial
. 1 . . . . . .2 . . . . . 3 . . . .4 `larr` (the number of terms is 4)

Here is more information about terms and polynomials
https://www.mathsisfun.com/algebra/polynomials.html