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
