a) \(\left(x+1\right)^3-3\left(x-1\right)^2=-2\)
\(\Rightarrow\left(x^3+3x+3x^2+1\right)-3\left(x^2+2x+1\right)=-2\)
\(\Rightarrow x^3+3x+3x^2+1-3x^2-6x-3=-2\)
\(\Rightarrow x^3-6x-2=-2\)
\(\Rightarrow x^3-6x=-2+2\)
\(\Rightarrow x^3-6x=0\)
\(\Rightarrow x\left(x^2-6\right)=0\)
\(\Rightarrow\left[{}\begin{matrix}x=0\\x^2-6=0\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=0\\x=\pm\sqrt{6}\end{matrix}\right.\)
b) \(\left(x-3\right)^2=x+\left(x+5\right)^2\)
\(\Rightarrow x^2-6x+9=x+x^2+10x+25\)
\(\Rightarrow x^2-6x-x^2-11x=25-9\)
\(\Rightarrow-17x=16\)
\(\Rightarrow x=-\dfrac{16}{17}\)
c) \(\left(x-8\right)\left(8+x\right)=3+\left(x+2\right)^2\)
\(\Rightarrow x^2-64=3+x^2+4x+4\)
\(\Rightarrow x^2-x^2-4x=3+4+64\)
\(\Rightarrow-4x=71\)
\(\Rightarrow x=-\dfrac{71}{4}\)
d) \(\left(x-3\right)^3=\left(x+2\right)\left(x^2-2x+4\right)-35\)
\(\Rightarrow x^3-9x^2+27x-27=x^3+8-35\)
\(\Rightarrow x^3-x^3-9x^2-27=-27\)
\(\Rightarrow-9x^2=-27+27\)
\(\Rightarrow-9x^2=0\)
\(\Rightarrow x^2=0\)
\(\Rightarrow x=0\)
