a monomial multiplied by a polynomial
This is best. Maybe.
It is important for students to understand the commutative property of multiplication (and other binary operators) and become procedurally fluent in using it with numbers, variables, monomials, polynomials and all other forms of algebraic expressions. However, after reading your comments carefully, I see what you really mean and I think there is an answer for it.
When we write a monomial, we try to follow the convention of writing numbers before letters. For example, by convention we write $4x$ not $x4$. Both are mathematically correct. But we are human and some ways a clearer than other ways. Also, using conventions reduces interpretive effort, allowing us to get more done in less time.
$4x^2+4x$ factorises to $4x(x+1)$. We prefer $4x(x+1)$ over $(x+1)4x$. Therefore, by convention, not mathematical rule, we put monomials before polynomials in a factorised expression.
That’s fine for mathematical expressions, but how about the word expression “a monomial multiplied by a polynomial”? Conventions get a little muddy here. If conventions get muddy, it is probably because there isn’t really a need for a convention. You could argue that the order in which things should appear in a sentence is the order in which they appear in the mathematical expression. So, for $4x$ we would say “$4$ multiplied by $x$” and not “$x$ multiplied by $4$”.
Some will argue that because $x$ multiplied by $4$ means $4$ acts on $x$, the phrase “multiplied by” is not commutative. However, when used in the context of mathematics, the binary multiplication operator is commutaive and the product of $4$ acting on $x$ is the same as the product of $x$ acting on $4$.
So, do you strictly apply the language syntax order that matches the desired conventional order in the mathematical expression?
I suggest not.
I have recently been reading Steven Pinker (The Sense of Style) and what he emphasises in writing is “flow”. Consider the flow of your writing. Not just a touchy-feely “flow” but an actual cognitive flow. If students are trying to visualise mathematical expressions as you speak or write, then it would be choppy to say B before A if you then by convention write A before B, unless they are so proficient with the syntax of English language that they would view this as an unforgivable travesty.
If you intend to write the monomial before the polynomial in a mathematical expression (e.g. $3x(x+5)$) then you should say $3x$ multiplied by $x+5$, or a monomial multiplied by a polynomial.
For the sake of consistency (and to avoid confusion with your students, especially those for whom English is not their native language), use the word order that matches the expression order – from left to right!