Đặt \(A=\left(x+3\right)^4+\left(x+1\right)^4-16\)

\(=\left\lbrack\left(x+2\right)+1\right\rbrack^4+\left\lbrack\left(x+2\right)-1\right\rbrack^4-16\)

Đặt b=x+2

=>\(A=\left(b+1\right)^4+\left(b-1\right)^4-16\)

\(=\left(b^2+2b+1\right)^2+\left(b^2-2b+1\right)^2-16\)

\(=\left(b^2+1\right)^2+4b\left(b^2+1\right)+4b^2+\left(b^2+1\right)^2-4b\left(b^2+1\right)+4b^2-16\)

\(=2\left(b^2+1\right)^2+8b^2-16\)

\(=2\left\lbrack\left(b^2+1\right)^2+4b^2-8\right\rbrack\)

\(=2\left\lbrack b^4+2b^2+1+4b^2-8\right\rbrack=2\left(b^4+6b^2-7\right)\)

\(=2\left(b^2+7\right)\left(b^2-1\right)=2\left(b^2+7\right)\left(b-1\right)\left(b+1\right)\)

\(=2\left\lbrack\left(x+2\right)^2+7\right\rbrack\left(x+2-1\right)\left(x+2+1\right)=2\left(x+1\right)\left(x+3\right)\left\lbrack\left(x+2\right)^2+7\right\rbrack\)

Đặt \(A=\left(x+3\right)^4+\left(x+1\right)^4-16\)

\(=\left\lbrack\left(x+2\right)+1\right\rbrack^4+\left\lbrack\left(x+2\right)-1\right\rbrack^4-16\)

Đặt b=x+2

=>\(A=\left(b+1\right)^4+\left(b-1\right)^4-16\)

\(=\left(b^2+2b+1\right)^2+\left(b^2-2b+1\right)^2-16\)

\(=\left(b^2+1\right)^2+4b\left(b^2+1\right)+4b^2+\left(b^2+1\right)^2-4b\left(b^2+1\right)+4b^2-16\)

\(=2\left(b^2+1\right)^2+8b^2-16\)

\(=2\left\lbrack\left(b^2+1\right)^2+4b^2-8\right\rbrack\)

\(=2\left\lbrack b^4+2b^2+1+4b^2-8\right\rbrack=2\left(b^4+6b^2-7\right)\)

\(=2\left(b^2+7\right)\left(b^2-1\right)=2\left(b^2+7\right)\left(b-1\right)\left(b+1\right)\)

\(=2\left\lbrack\left(x+2\right)^2+7\right\rbrack\left(x+2-1\right)\left(x+2+1\right)=2\left(x+1\right)\left(x+3\right)\left\lbrack\left(x+2\right)^2+7\right\rbrack\)

\(x^4+3x^3+12x-16\)

\(=x^4+4x^3+4x^2+16x-x^3-4x^2-4x-16\)

\(=x\left(x^3+4x^2+4x+16\right)-\left(x^3+4x^2+4x+16\right)\)

\(=\left(x-1\right)\left(x^3+4x^2+4x+16\right)\)

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

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

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

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

Đặt b=x+4

=>\(A=\left(b-1\right)^4+\left(b+1\right)^4-2\)

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

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

\(=2\left(b^2+1\right)^2+8b^2-2\)

\(=2\left\lbrack\left(b^2+1\right)^2+4b^2-1\right\rbrack\)

\(=2\cdot\left\lbrack b^4+2b^2+1+4b^2-1\right\rbrack=2\left(b^4+6b^2\right)=2b^2\left(b^2+6\right)\)

\(=2\left(x+4\right)^2\left\lbrack\left(x+4\right)^2+6\right\rbrack\)

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

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

Đặt b=x+4

=>\(A=\left(b-1\right)^4+\left(b+1\right)^4-2\)

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

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

\(=2\left(b^2+1\right)^2+8b^2-2\)

\(=2\left\lbrack\left(b^2+1\right)^2+4b^2-1\right\rbrack\)

\(=2\cdot\left\lbrack b^4+2b^2+1+4b^2-1\right\rbrack=2\left(b^4+6b^2\right)=2b^2\left(b^2+6\right)\)

\(=2\left(x+4\right)^2\left\lbrack\left(x+4\right)^2+6\right\rbrack\)

Đặt đúng môn học Toán bạn nhé. -

Ta có \(x^4+10x^3+32x^2+40x+16=\left(x^4+2x^3\right)+\left(8x^3+16x^2\right)+\left(16x^2+32x\right)+\left(8x+16\right)\)

\(=x^3\left(x+2\right)+8x^2\left(x+2\right)+16x\left(x+2\right)+8\left(x+2\right)\)

\(=\left(x+2\right)\left(x^3+8x^2+16x+8\right)=\left(x+2\right)\left(x+2\right)\left(x^2+6x+4\right)\)

\(=\left(x+2\right)^2\left(x^2+6x+4\right)\)