### Theory:

Multiplying monomial with a polynomial:

Let us recall the distributive proportion.
If $$a$$ is a constant, $$x$$ and $$y$$ are variables, then $$a(x + y) = ax + ay$$.
Example:
1. Suppose there will be $$x$$ number of bags and a bag contains $$3$$ cupcakes of '$$p$$' packs, $$7$$ chocolates of '$$q$$' packs and $$5$$ cookies of '$$r$$' packs. The total number of items can be identified by adding the number of items in the bag and product with the number of bags.

This can be written as $x\left(3p+7q+5r\right)$.

Applying the distributive property,

$$= 3px + 7qx + 5rx$$.

2. Find the product of $3{p}^{3}q$ and $\left(2{\mathit{pq}}^{3}-5{p}^{2}+3{q}^{4}\right)$.

$=\phantom{\rule{0.147em}{0ex}}3{p}^{3}q×\left(2{\mathit{pq}}^{3}-5{p}^{2}+3{q}^{4}\right)$

Applying the distributive property,

$$=$$ $$(3×2)$$ $$p^{3+1}q^{1+3}$$ $$+$$ $$(3×-5)$$$$p^{3+2}q$$ $$+$$ $$(3×3)p^3q^{1+4}$$

$$=$$ $$6p^4q^4-15p^5q+9p^3q^5$$.