If $x^4-3x^2-1=0$, then the value of $\left(x^6-3x^2+\frac{3}{x^2}-\frac{1}{x^6}+1\right)$ is:

Given equation:
$$x^4-3x^2-1=0$$
Divide the entire equation by $x^2$:
$$x^2-3-\frac{1}{x^2}=0⟹x^2-\frac{1}{x^2}=3$$
Now, cube both sides of the equation:
$${\left(x^2-\frac{1}{x^2}\right)}^3=3^3$$
$$x^6-\frac{1}{x^6}-3\left(x^2\right)\left(\frac{1}{x^2}\right)\left(x^2-\frac{1}{x^2}\right)=27$$
$$x^6-\frac{1}{x^6}-3(3)=27$$
$$x^6-\frac{1}{x^6}-9=27⟹x^6-\frac{1}{x^6}=36$$
The expression we need to evaluate is:
$$E=x^6-3x^2+\frac{3}{x^2}-\frac{1}{x^6}+1$$
Rearranging the terms:
$$E=\left(x^6-\frac{1}{x^6}\right)-3\left(x^2-\frac{1}{x^2}\right)+1$$
Substitute the known values into the expression:
$$E=36-3(3)+1$$
$$E=36-9+1=28$$