Find the roots of the function f(x) = (2x − 1)*(x^2 + 2x − 3), with x in R

First find the factors of ${\displaystyle (x^2+2x-3).}$
Compare the ${\displaystyle a{x}^2+bx+c}$ and ${\displaystyle x^2+2x-3}$, implies that $a=1,b=2$, and $c=-3$.
The ac-method of factoring quadratics: To find factors of quadratic ${\displaystyle a{x}^2+bx+c}$, Find two numbers whose sum is b and product is $a\cdot c$.
Here $a\cdot c=1\cdot (-3)=-3$and $b=2$.
To find factors of quadratic ${\displaystyle x^2+2x-3}$, find two numbers whose sum is 2 and the product is -3.
Such numbers are 3 and -1.
The factors of ${\displaystyle (x^2+2x-3)}$ are $(x+3)(x-1).$
The given function ${\displaystyle f\left(x\right)=(2x-1)\cdot (x^2+2x-3)}$ becomes
${\displaystyle f\left(x\right)=(2x-1)·(x+3)·(x-1)}$
Set $f(x)=0$, implies that
${\displaystyle (2x-1)\cdot (x+3)\cdot (x-1)=0}$
By using zero product property,
${\displaystyle (2x-1)=0,(x+3)=0,(x-1)=0}$
Implies that,
$2x=1,x=-3,x=1$
That is, the roots of the given function are,
${\displaystyle x=\frac{1}{2},x=-3,x=1.}$