How do you simplify \[\left( -5ab{{c}^{4}} \right)\left( -3{{a}^{3}}{{c}^{2}} \right)\left( -4{{a}^{2}}{{b}^{4}}{{c}^{3}} \right)\] ?

Hint: Problems like these are quite easy to solve and are simple to implement once we get the underlying concept behind the sun. For this particular problem we need to have some basic as well as advanced knowledge of factorisation, factor theorem and algebraic multiplication of different algebraic expressions. In this problem we are given three unknown parameters or variables which are considered under multiplication. We must remember one thing that when we multiply variables of the same type their powers get added and when we divide two variables of the same type then their power gets subtracted. Here we also need to take out the negative signs from the problem and when we take out two negative signs we get a positive sign and when we take out a negative and a positive sign the resultant sign becomes negative.

Complete step by step answer:
Now we start off with the solution to the given problem by writing that,
We can rearrange the given problem by taking out the three negative signs out of the bracket. The resultant sign becomes negative.
\[\Rightarrow \left( -5ab{{c}^{4}} \right)\left( -3{{a}^{3}}{{c}^{2}} \right)\left( -4{{a}^{2}}{{b}^{4}}{{c}^{3}} \right)=-\left( 5ab{{c}^{4}} \right)\left( 3{{a}^{3}}{{c}^{2}} \right)\left( 4{{a}^{2}}{{b}^{4}}{{c}^{3}} \right)\]
Now we multiply the terms and add the powers of the like terms to get,
\[\Rightarrow \left( -5ab{{c}^{4}} \right)\left( -3{{a}^{3}}{{c}^{2}} \right)\left( -4{{a}^{2}}{{b}^{4}}{{c}^{3}} \right)=-60{{a}^{1+3+2}}{{b}^{1+0+4}}{{c}^{4+2+3}}\]
Adding the terms of the powers we get the final result as,
\[\Rightarrow \left( -5ab{{c}^{4}} \right)\left( -3{{a}^{3}}{{c}^{2}} \right)\left( -4{{a}^{2}}{{b}^{4}}{{c}^{3}} \right)=-60{{a}^{6}}{{b}^{5}}{{c}^{9}}\]

Thus, our answer to the problem is \[-60{{a}^{6}}{{b}^{5}}{{c}^{9}}\].

Note: We need to have a clear-cut understanding of the concepts of factorisation and algebraic multiplications of different algebraic expressions. We need to be careful while multiplying the like terms in this problem, because the power of the like terms gets added and while dividing gets subtracted. We also need to remember the concept that two negative signs on multiplication gives a positive sign and one positive and negative sign on multiplication gives a negative sign.