Answer
$(a)$ Symmetry with respect to the origin
$(b)$ $x$-intercept $A(0,1)$ and $B(0,-1)$
$y$-intercept $C(-1,0)$ and $D(1,0)$
Work Step by Step
$(a)$ As we can see from the image of the graph above, it is symmetrical with respect to the origin.
Algebraically, if we replace $x$ by $-x$ and $y$ by $-y$ equation will be still equivalent to the original one.
$(b)$
$x$-intercept happens when $y=0$
$x^2+4x\times0+0^2=1$
$x^2=1$
$x=±1$
$A(0,1)$ and $B(0,-1)$
$y$-intercept happens when $x=0$
$0^2+4y\times0+y^2=1$
$y^2=1$
$y=±1$
$C(-1,0)$ and $D(1,0)$