Đáp án+Giải thích các bước giải:
$1. x^2+x+1\\ =x^2+x+\dfrac{1}{4}+\dfrac{3}{4}\\ =\left(x+\dfrac{1}{2}\right)^2+\dfrac{3}{4} >0 \ \forall \ x\\ 3. x-x^2-1\\ =-x^2+x-1\\ =-x^2+x-\dfrac{1}{4}-\dfrac{3}{4}\\ =-\left(x-\dfrac{1}{2}\right)^2-\dfrac{3}{4} <0 \ \forall \ x\\ 4. 4x-x^2-5\\ =-x^2+4x-5\\ =-x^2+4x-4-1\\ =-(x-2)^2-1<0 <0 \ \forall \ x\\ 5. -x^2 + 10x- 35 \\ = -x^2 + 10x- 25 -10\\ =-(x-5)^2-10 <0 \ \forall \ x$