The number of integral values of m for which the equation 1+m2x2−2(1+3m)x+(1+8m)=0 has no real roots is

Let D be the discriminant of the given equationThen,D=4(1+3m)2−41+m2(1+8m)⇒ D=49m2+6m+1−8m3−m2−8m−1⇒ D=4−8m3+8m2−2m⇒ D=−8m4m2−4m+1=−8m(2m−1)2If the given equation has no real roots, thenD<0⇒−8m(2m−1)2<0⇒m>0 and m≠12Hence, m can take infinitely many integral values.