The general strategy for factoring an expression with 4 terms is to split the terms into the odd and even powers of the variable, if possible.
15x^3 + 10x^2 + 3x + 2 = 15x^3 + 3x + 10x^2 + 2
this is just an application of the commutative law of addition: a+ b = b + a.
The greatest common factor of 15x^3 + 3x is 3x, so
15x^3 + 3x = 3x ( 5x^2 + 1 )
The greatest common factor of 10x^2 + 2 is 2, so
10x^2 + 2 = 2 ( 5x^2 + 1 ).
Noting that 5x^2 + 1 is a common factor of both of these expressions,
15x^3 + 3x + 10x^2 + 2 = ( 3x + 2 ) ( 5x^2 + 1 )
(This may not always work, but you should try this method of factoring just in case.)
If you are limited to the real number system, that is all you can do to factor this. However, in the complex number system, you can also factor 5x^2 + 1 as the sum of two squares:
5x^2 + 1 = ( x√5 + i ) ( x√5 - i ) where i = √-1.
So, if complex factoring is allowed:
( 3x + 2 ) ( x√5 + i ) ( x√5 - i )
is the completely factored form.
