If `(x^2+x=2)62=(a-3)(x^2+x+1)(x^2+x+2)+(a-4)(x^2+x+1)^2=0` has at least one root, then find the complete set of values of `adot`

Explanation

Let `t = x^(2) + x + 1 rArr r in [(3)/(4), infty)`
Hence,
`(t+1)^(2) - (a -3)t(t+ 1) + (a - 4)t^(2)= 0`
or `t^(2)+2t + 1 - (a -3)(t^(2)+ 1) + (a - 4)t^(2)= 0`
or ` t(2 - a + 3) + 1 = `
or `t= (1) /(a -5)`
`rArr (1)/(a -5) ge (3)/(4)`
`rArr (19 - 3a)/((a - 5)) gt 0`
`rArr a in(5, (19)/(3)]`