IF a,b,c are real numbers and 3b^2-8ac<0,then the equation ax^4+bx^3+cx^2+5x+7=0 cannot have all roots real. prove this.

The wording of the question is a dead giveaway.

Suppose all the roots r₁, r₂, r₃ and r₄ are real, then we must have

so that

Now, from Vieta's formulas, . So, if all roots are real, we must necessarily have  

or .

Since we are given that  it implies that all roots cannot be real.