How do we find the product $2p{r^2}(2pr + 5{p^2}r - 15p)$ ?

Hint: For solving this particular problem , we have to multiply the monomial present outside the parentheses that is $2p{r^2}$ , with all the terms present inside the parentheses that is $2pr + 5{p^2}r - 15p$ .

Complete step-by-step solution:
We have given ,
$2p{r^2}(2pr + 5{p^2}r - 15p)$
We have to find the product ; this works exactly the same way as we use to do with the numbers. For numbers, if it is given $a(b + c)$ it is equals to $ab + ac$, similarly if we have a monomial and we have to multiply that monomial by a polynomial, we will follow the same rule ,
Therefore, we will get,
$ \Rightarrow 2p{r^2}(2pr + 5{p^2}r - 15p)$
$ \Rightarrow 2p{r^2} \times 2pr + 2p{r^2} \times 5{p^2}r + 2p{r^2} \times ( - 15p)$
Just remember two things. First, two negative numbers will always give a positive number. Second, if we multiply only one negative value with the positive value , our answer will also be negative always . Just remember these two rules and the rest is simple multiplication only.
$ \Rightarrow 4{p^2}{r^3} + 10{p^3}{r^3} - 30{p^2}{r^2}$
we get the required result.

Hence the correct answer is $4{p^2}{r^3} + 10{p^3}{r^3} - 30{p^2}{r^2}$ .

Note: Just remember two things. First, two negative numbers will always give a positive number. Second, if we multiply only one negative value with the positive value , our answer will also be negative always . Just remember these two rules and the rest is simple multiplication only. In order to avoid confusion and to make sure that everybody always arrives at the identical result, mathematicians established a typical order of operations for calculations that involve over one mathematical operation. Arithmetic operations should be dole out within the following order:
1. We must Simplify the given expressions inside parentheses ( ), brackets [ ], braces and fractions bars first.
2. Then we have to evaluate all the powers present in the given expression.
3. Then we have to do all multiplications part first and then the divisions part from left to right direction.
4. Then we have to do all additions and then the subtractions part from left to right direction.