How do you subtract $(3{x^3} - 4{x^2} + 6x) - \left( {5{x^3} + 2{x^2} - 3x} \right)?$

Hint: In this question we have to subtract the polynomials. As we know that subtraction is an arithmetic operation which is performed on a collection of objects by removing a certain number of objects. In this question, for subtracting one polynomial from the other, we first reverse the signs of the terms in the polynomial that is to be subtracted, then we rearrange the terms so that the same variable terms gets grouped together and after that we will do the arithmetic operations on the coefficients of similar terms, like $8{x^3} - 5{x^3} = 3{x^3}$.

Complete step by step solution:
In this question we have polynomials, $(3{x^3} - 4{x^2} + 6x) - \left( {5{x^3} + 2{x^2} - 3x} \right)$.
By applying the above method, we will first reverse the signs of the terms which is to be subtracted i.e. $\Rightarrow 3{x^3} - 4{x^2} + 6x - 5{x^3} - 2{x^2} + 3x$.
This is the new expression, now grouping the similar terms and subtracting them we have:
$\Rightarrow3{x^3} - 5{x^3} - 4{x^2} - 2{x^2} + 3x + 6x = - 2{x^3} - 6{x^2} + 9x$.

Hence the required answer is $ - 2{x^3} - 6{x^2} + 9x$.

Note: Before solving this kind of questions, we should be familiar with all the arithmetic operations and their definitions. It is important to understand how these operations work on polynomials, We should note that addition or subtraction of a polynomial from another polynomial will give the same degree of polynomial in result unless the highest degree coefficients get cancelled with each other.