If the discriminant of a quadratic equation #ax^2+bx+c=0#, which is #Delta=b^2-4ac# is a complete square of a rational number, we can solve it by factorization.
Here in the given equation #21x^2+22x-24=0#, #Delta=22^2-4xx21xx(-24)=484+2016=2500=50^2#, hence we can solve it by factorization.
For factorization, split middle term #b=22# in two parts whose product is #ac=-21xx24=-504#. In other words two numbers, whose difference is #22# and product is #504#. These are are #-14# and #36#.
Therefore, #21x^2+22x-24=0# can be written as
#21x^2-14x+36x-24=0#
or #7x(3x-2)+12(3x-2)=0#
or #(7x+12)(3x-2)=0#
Hence either #7x+12=0# i.e. #x=-12/7#
or #3x-2=0# i.e. #x=2/3#
