What is Multiplication of Monomials by Polynomials?

It works the same as with numbers. For numbers, you know that a(b+c) equals ab+ac.
For the same reason, if you have a monomial and you want to multiplicate it by a polynomial (which is a sum of monomials with some coefficients!), you follow the same rule.
For example, if your monomial is ${\displaystyle 3x^2}$, and your polynomial is ${\displaystyle 3+2x-5x^2+8x^3}$, the product is
${\displaystyle 3x^2(3+2x-5x^2)+8x^3}$
you will calculate is as
${\displaystyle 3x^2\cdot 3+3x^2\cdot 2x-3x^2\cdot 5x^2+3x^2\cdot 8x^3}$, which is
${\displaystyle 9x^2+6x^3-15x^4+24x^5}$

When multiplying polynomials, each term from one polynomial must be multiplied by each term of the other polynomial. Since each term in a polynomial is a monomial, multiplying polynomials becomes multiplying monomials. ... The factors are regrouped, and then multiplied.