Quadratics by factoring 2z(5z-2)=-5z+2

Simplify the quadratic equation:
${\displaystyle 10z^2-4z=-5z+2}$
${\displaystyle 1-z^2-4z+5z-2=0}$
${\displaystyle 10z^2+z-2=0}$
Factorize the quadratic equation
${\displaystyle 10z^2+5z-4z-2=0}$
${\displaystyle 5z(2z+1)-1(2z+1)=0}$
${\displaystyle (5z-1)(2z+1)}$
Use the principle of zero products which states that if the product of two numbers is zero then at least one of the factors is 0.
${\displaystyle (5z-1)=0\text{or}(2z+1)=0}$
${\displaystyle z-\frac{1}{5}}$ or ${\displaystyle z=-\frac{1}{2}}$