Answer
There are no x-intercepts.
The y-intercept is (0,1)
The equation has symmetry only with respect to the y-axis.
Work Step by Step
To find the x-intercept(s), we set y to 0 and solve for x:
$0=x^4+2x^2+1$
$0=(x^2+1)^2$
$0=x^2+1$
$x^2=-1$
$\sqrt{x^2}=\sqrt{-1}$
It is not possible to take a square root of a negative number. That means this equation has no x-intercepts.
To find the y-intercept(s), we set x to y and solve for y:
$y=0^4+2(0)^2+1$
$y=1$
To test for symmetry with respect to the x-axis, we substitute y for -y and check if it equals the original equation:
$(-y)=x^4+2x^2+1$
$-y=x^4+2x^2+1$ nope
To test for symmetry with respect to the y-axis, we substitute x for -x and check if it equals the original equation:
$y=(-x)^4+2(-x)^2+1$
$y=x^4+2x^2+1\checkmark$
To test for symmetry with respect to the origin, we substitute x for -x, substitute y for -y and check if it equals the original equation:
$(-y)=(-x)^4+2(-x)^2+1$
$-y=x^4+2x^2+1$ nope