Đặt y = \(x+1=\sqrt[3]{8+2\sqrt{14}}+\sqrt[3]{8-2\sqrt{14}}\)
=> \(y^3=8+2\sqrt{14}+8-2\sqrt{14}+3\sqrt[3]{\left(8+2\sqrt{14}\right)\left(8-2\sqrt{14}\right)}.y\)
<=> \(y^3=16+6y\)
=> \(\left(x+1\right)^3=16+6\left(x+1\right)\)
=> \(x^3+3x^2+3x+1=6x+32\)
<=> \(x^3+3x^2-3x-5=26\)
Ta có:
\(x^6+3x^5-3x^4-2x^3+9x^2-9x+2018\)
= \(x^6+3x^5-3x^4-5x^3+3x^3+9x^2-9x-15+2033\)
= \(\left(x^3+3x^2-3x-5\right)\left(x^3+3\right)+2033\)
= \(26x^3+2111\)
\(=26\left(\sqrt[8]{8+2\sqrt{14}}+\sqrt[8]{8-2\sqrt{14}}-1\right)^3+2033\)
