Factorise polynomial $\left( {3{x^2} - 5x - 2} \right)$ .

Hint: Given polynomial is of degree 2. Polynomials of degree 2 are known as Quadratic polynomials. Quadratic polynomials can be factored by the help of splitting the middle term method. In this method, the middle term is split into two terms in such a way that the polynomial remains unchanged.

Complete step-by-step solution:
For factorising the given quadratic polynomial $\left( {3{x^2} - 5x - 2} \right)$ , we can use the splitting method in which the middle term is split into two terms such that the sum of the terms gives us the original middle term and product of the terms gives us the product of the constant term and coefficient of ${x^2}$.
So, $\left( {3{x^2} - 5x - 2} \right)$
$ = $$3{x^2} - \left( {6 - 1} \right)x - 2$
We split the middle term $ - 5x$ into two terms $ - 6x$ and $x$ since the product of these terms, $ - 6{x^2}$ is equal to the product of the constant term and coefficient of ${x^2}$ and sum of these terms gives us the original middle term, $ - 5x$.
$ = $$3{x^2} - 6x + x - 2$
$ = $\[3x\left( {x - 2} \right) + \left( {x - 2} \right)\]
$ = $\[\left( {3x + 1} \right)\left( {x - 2} \right)\]
So, the factored form of the quadratic polynomial $\left( {3{x^2} - 5x - 2} \right)$ is \[\left( {3x + 1} \right)\left( {x - 2} \right)\].

Note: Splitting of the middle term can be a tedious process at times when the product of the constant term and coefficient of ${x^2}$ is a large number with a large number of divisors. Special care should be taken in such cases. Similar to quadratic polynomials, quadratic equations can also be solved using factorisation methods. Besides factorisation, there are various methods to solve quadratic equations such as completing the square method and using the Quadratic formula.