How do you factor completely $ 12a{x^3} + 20b{x^2} + 32cx $ ?

Hint: The factorization of polynomial expression is done by clubbing all the values of a given variable in a polynomial expression. The polynomial expression is then expressed in the form of a product of the value which all the terms that are added contain for example in the above given polynomial expression we see that a constant $ 4 $ is part of all the terms in the given polynomial expression as it is a factor of $ 12 $ along with being a factor of $ 20\,,{\text{32}} $ . Thus $ 4 $ is one factor of the expression similarly we can see that a variable $ x $ is part of all the terms that are present in the given polynomial expression.

Complete step by step solution:
The first term contains power $ 3 $ of the variable while the other terms in the polynomial expression contain power $ 2 $ and $ 1 $ respectively. But since the least power present here is one the factor will only contain power $ 1 $ of the variable in question. Thus we will create the multiplication expression using the common variables or constant present in all the terms.
The following factorization can be solved by taking out the common from all the terms in the given polynomial expression from the hint we can write that $ 4x $ is common to the all the parts of the polynomial so we will write:
 $ 12a{x^3} + 20b{x^2} + 32cx = 4x\left( {3a{x^2} + 5bx + 8c} \right) $
Thus our given polynomial is solved.
So, the correct answer is “ $ 4x\left( {3a{x^2} + 5bx + 8c} \right) $ ”.

Note: When finding the factors of a polynomial expression always try to find the constants and variables that are common in all the terms of that expression then take out those common elements and then write them outside as factors of that polynomial.