a: \(P\left(x\right)=2x^3-x^4+1+2x^2+5x^4-x^3\)

\(=2x^3-x^3+\left(5x^4-x^4\right)+2x^2+1\)

\(=4x^4+x^3+2x^2+1\)

\(Q\left(x\right)=-3x^4-1+5x^3-x^2-6x^2-4x^3\)

\(=-3x^4+\left(5x^3-4x^3\right)+\left(-x^2-6x^2\right)-1\)

\(=-3x^4+x^3-7x^2-1\)

b: \(P\left(-2\right)=4\cdot\left(-2\right)^4+\left(-2\right)^3+2\cdot\left(-2\right)^2+1\)

\(=4\cdot16-8+8+1\)

=64+1

=65

c: P(x)+Q(x)

\(=4x^4+x^3+2x^2+1-3x^4+x^3-7x^2-1=x^4+2x^3-5x^2\)

d: Q(x)-P(x)

\(=-3x^4+x^3-7x^2-1-4x^4-x^3-2x^2-1=-7x^4-9x^2-2\le-2<0\forall x\)

a: \(P\left(x\right)=2x^3-x^4+1+2x^2+5x^4-x^3\)

\(=2x^3-x^3+\left(5x^4-x^4\right)+2x^2+1\)

\(=4x^4+x^3+2x^2+1\)

\(Q\left(x\right)=-3x^4-1+5x^3-x^2-6x^2-4x^3\)

\(=-3x^4+\left(5x^3-4x^3\right)+\left(-x^2-6x^2\right)-1\)

\(=-3x^4+x^3-7x^2-1\)

b: \(P\left(-2\right)=4\cdot\left(-2\right)^4+\left(-2\right)^3+2\cdot\left(-2\right)^2+1\)

\(=4\cdot16-8+8+1\)

=64+1

=65

c: P(x)+Q(x)

\(=4x^4+x^3+2x^2+1-3x^4+x^3-7x^2-1=x^4+2x^3-5x^2\)

d: Q(x)-P(x)

\(=-3x^4+x^3-7x^2-1-4x^4-x^3-2x^2-1=-7x^4-9x^2-2\le-2<0\forall x\)

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

a: Ta có: \(x^4-2x^3+2x-1\)

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

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

\(=\left(x-1\right)^3\cdot\left(x+1\right)\)

b: Ta có: \(-a^4+a^3+2a^3+2a^2\)

\(=-a^2\left(a^2-a-2a-2\right)\)

c: Ta có: \(x^4+x^3+2x^2+x+1\)

\(=x^4+x^3+x^2+x^2+x+1\)

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

\(=x^3\left(x+2\right)-x\left(x+2\right)\)

\(=\left(x+2\right)\cdot x\cdot\left(x+1\right)\left(x-1\right)\)

Vì đây là tích của bốn số nguyên liên tiếp

nên \(\left(x+2\right)\cdot x\cdot\left(x+1\right)\cdot\left(x-1\right)⋮24\)

Chọn B

\(a,x^4-2x^3+6x^2+x+14\\ =\left(x^4-3x^3+7x^2\right)+\left(x^3-3x^2+7x\right)+\left(2x^2-6x+14\right)\\ =\left(x^2-3x+7\right)\left(x^2+x+2\right):\left(x^2-3x+7\right)=x^2+x+2\)

Ta có \(x^2+x+2=x^2+x+\dfrac{1}{4}+\dfrac{7}{4}=\left(x+\dfrac{1}{2}\right)^2+\dfrac{7}{4}\ge\dfrac{7}{4}>0\)

Vậy ...

\(b,A=x^3+3xy+y^3\\ A=\left(x+y\right)\left(x^2-xy+y^2\right)+3xy\\ A=x^2-xy+y^2+3xy\\ A=x^2+2xy+y^2=\left(x+y\right)^2=1\)