How do you simplify \[\left( 4{{p}^{2}}q+5{{p}^{3}}{{q}^{2}}-6pq \right)-\left( -2{{p}^{3}}{{q}^{2}}+5pq-8{{p}^{2}}q \right)\] ?

Hint: For problems like these we need to have a fair amount of ideas regarding factorization and associative property. In the rule of factorization, we need to factorize any given equation following a set of rules. In this above given problem, we first need to multiply the negative sign as given in the second term of the equation and we must remember that all the signs inside the first bracket get reversed once we multiply the inside terms with a negative number. After all the signs in the problem have been fixed, we need to add or subtract the remaining terms in a simple algebraic manner.

Complete step by step solution:
Now we start off with the solution of the given problem by writing that, we first of all multiply the negative number inside the bracket and convert it into a positive sign as,
\[\left( 4{{p}^{2}}q+5{{p}^{3}}{{q}^{2}}-6pq \right)+\left( 2{{p}^{3}}{{q}^{2}}-5pq+8{{p}^{2}}q \right)\]
Now, we remove the brackets from the given equation and write it down as,
\[=4{{p}^{2}}q+5{{p}^{3}}{{q}^{2}}-6pq+2{{p}^{3}}{{q}^{2}}-5pq+8{{p}^{2}}q\]
We now, add and subtract the like terms accordingly and then further evaluate the value and write,
\[=12{{p}^{2}}q+7{{p}^{3}}{{q}^{2}}-11pq\]
Now, we can clearly see that the factor\[pq\] is present in each and every term, so taking this common from all the terms, we can hence write it as,
 \[=pq\left( 12p+7{{p}^{2}}q-11 \right)\]
From the terms inside the bracket, we see that the factor \[p\] is present in the first and the second terms, so taking this common, we can write,
 \[=pq\left\{ p\left( 12+7pq \right)-11 \right\}\]
We can see from these that the equation cannot be simplified further. So this will be our final answer.

Note:
For such types of problems, we must remember the rules of factorization and associative properties. We must also remember the law, that when we multiply a term with a negative number the sign gets reversed, while if we multiply it by a positive number the sign remains unchanged. We should also take the factors common very carefully and check whether the given equation can be simplified further or not.