a: Ta có: \(-x^2+4x-5\)

\(=-\left(x^2-4x+5\right)\)

\(=-\left(x^2-4x+4+1\right)\)

\(=-\left(x-2\right)^2-1< 0\forall x\)

b: Ta có: \(x^4\ge0\forall x\)

\(3x^2\ge0\forall x\)

Do đó: \(x^4+3x^2\ge0\forall x\)

\(\Leftrightarrow x^4+3x^2+3>0\forall x\)

c: Ta có: \(\left(x^2+2x+3\right)=\left(x+1\right)^2+2>0\forall x\)

\(x^2+2x+4=\left(x+1\right)^2+3>0\forall x\)

Do đó: \(\left(x^2+2x+3\right)\left(x^2+2x+4\right)>0\forall x\)

\(\Leftrightarrow\left(x^2+2x+3\right)\left(x^2+2x+4\right)+3>0\forall x\)​

b: Ta có: \(x^4\ge0\forall x\)

\(3x^2\ge0\forall x\)

Do đó: \(x^4+3x^2\ge0\forall x\)

\(\Leftrightarrow x^4+3x^2+3>0\forall x\)

c: Ta có: \(\left(x^2+2x+3\right)=\left(x+1\right)^2+2>0\forall x\)

\(x^2+2x+4=\left(x+1\right)^2+3>0\forall x\)

Do đó: \(\left(x^2+2x+3\right)\left(x^2+2x+4\right)>0\forall x\)

\(\Leftrightarrow\left(x^2+2x+3\right)\left(x^2+2x+4\right)+3>0\forall x\)

x^2 + 2x + 2 = x^2 + 2.x.1 + 1^2 +1 = (x + 1)^2 + 1 > 0

-x^2 + 4x - 4 = -(x^2 - 2.x.2 + 2^2) = -(x - 2)^2 <= 0

a) ta co ; x^2+ 2x+ 2= (x²+2x+1)+1=(x+1)²+1>0

vi (x+1)²>hoặc=0;1>0suy ra x^2+ 2x+ 2>0

b)ta co  -x²+4x-4=-(x²-4x+4)=-(x-2)²<0

a) x^2 + 2x + 2 = ( x^2 + 2x +1 ) + 1 =( x + 1)^2 +1 >0 với mọi x 

b) -x^2 + 4x - 4 = -( x^2 -4x + 4 ) = - ( x - 2)^2  ≤ 0 với mọi x

2x^2+4x+3=2(x^2+2x+1)+1=2(x+1)^2+1>0 với mọi x

2x²+4x+3=2(x²+2x+3/2)=2(x²+2x+1+1/2)=2(x+1)²+1>0 với mọi x

2x^2+4x+2+1>0

2(x+1)^2+1>0 (đúng) 

suy ra đpcm 

\(2x^2+4x+3\)

\(=2\left(x^2+2x+\frac{3}{2}\right)\)

\(=2\left(x^2+2x+1^2-1^2+\frac{3}{2}\right)\)

\(=2\left[\left(x+1\right)^2+\frac{1}{2}\right]\)

\(=2\left(x+1\right)^2+1>0\forall x\)

a) Xét \(x^2-4x+4=\left(x-2\right)^2\ge0\)

<=> \(x^2-4x\ge-4>-5\)

b) \(2x^2+4y^2-4x-4xy+5\)

= \(\left(x^2-4x+4\right)+\left(x^2-4xy+4y^2\right)+1\)

= \(\left(x-2\right)^2+\left(x-2y\right)^2+1\ge1>0\)

I not sure for this answer if have any trouble you can ask me

a)\(\sqrt{x^2-4x+5}\ge\forall x\)

\(\Leftrightarrow\sqrt{x^2-4x+4+1}\)

\(\Leftrightarrow\sqrt{\left(x+1\right)}^2+1\)

mà \(\sqrt{\left(x+1\right)^2}\ge0\forall x\)

nên \(\sqrt{\left(x+1\right)^2}+1>0\forall x\)

câu a chưa rõ đề, bắt chứng minh nhưng ko biết \(\ge\) cái j ms đc chứ ạ ?

Ta có : x² + 2x + 2

= x² + 2x + 1 + 1

= (x + 1)² + 1 \(\ge1\forall x\)

Vậy  x² + 2x + 2 \(>0\forall x\)

Ta có : x² + 2x + 2

=> x² + 2x + 1 + 1

=> ( x + 1)² + 1  >  1\(\forall x\)

Vậy x² + 2x + 2   > \(0\forall x\)

( Câu trả lời bằng hình ảnh minh họa )