To factorize #ax^2+bx+c#, one needs to first check about discriminant #b^2-4ac#, which in #3x^2−15x+16# is #(-15)^2-4xx3xx16=225-192=33#. As it is not the square of a rational number, we will not have binomials with rational coefficients as factors.
Hence, to find factors let us solve the equation #3x^2−15x+16=0# using quadratic formula which gives solution of #ax^2+bx+c=0#
as #x=(-b+-sqrt(b^2-4ac))/(2a)#.
Hence solution of #3x^2−15x+16=0# is #x=(-(-15)+-sqrt((-15)^2-4xx3xx16))/(2xx3)# or
#x=(15+-sqrt33)/6#
Hence factors are #3(x-(15+sqrt33)/6)(x-(15-sqrt33)/6)#
We have multiplied by #3# as coefficient of #x^2# is #3#.
