$Since,\ f\left(z\right)is\ analytic,\ Cauchy^{'}s\ Reimann^{'}s \[2ex]equation\ will\ be\ satisfied.$ $\displaystyle (1)If\ x\not=0\ and\ y\not=0\ then\ ux,\ uy,\ vx,\ vy\ , are\ continuous\ function\ of\ x\ \&\ y. \[2ex]

\displaystyle (2)\ ux\ =vy\ \ \ \ \ \ \ \ \ \ \ \ and \[2ex]

\displaystyle (3)\ uy\ =-\ vx\ \[2ex]

\displaystyle z=x+iy \[2ex]

$ $\displaystyle f\left(z\right)=u+iv=\ \left(ax^4+bx^2y^2+cy^4+dx^2-2y^2\right)+i\ \left(4x^3\ y-exy^3+4xy\right) $ $\displaystyle Comparing\ real\ and\ imaginary\ parts \[4ex]

\displaystyle u=\left(ax^4+bx^2y^2+cy^4+dx^2-2y^2\right)\ and\ \[3ex] \displaystyle v=\left(4x^3\ y-exy^3+4xy\right)\ \ \ \ \ \ \ \ \ \ \[2ex]

\displaystyle \therefore{}\ ux=4ax^3+2bxy^2+2dx\ \ \[2ex]

$ $\displaystyle uy=2bx^2y+4cy^3-4y\ \ \ \ \[2ex]

\displaystyle vx\ =12x^2y-ey^3+4y\ \[2ex]

\displaystyle

\[2ex]

\displaystyle vy=4x^3-3exy^2+4x\ \ \ \ \ \ \ \ \ \ \[2ex]

\displaystyle from\ \left(2\right)and\ \left(3\right)we\ have\ u_x=v_y,\ u_y=\ -v_x\ \[2ex]

\displaystyle 4ax^3+2bxy^2+2dx=\ 4x^3-3exy^2+4x\ \ \[2ex]

\displaystyle 2bx^2y+4cy^3-4y\ \ \ =\ -(12x^2y-ey^3+4y)\ \[2ex]

$ $Comparing \ the \ corresponding \ co-efficients,\[2ex]

$ $\displaystyle 4a=4\ ,\ a=1 \[2ex]

\displaystyle 2d=4,\ d=2\ \[2ex]

\displaystyle 2b=-12\ ,\ b=\ -6\ \[2ex]

\displaystyle 2b=-3e=\ -12\ ,\ e=4 \[2ex]

\displaystyle 4c=e,\ c=\frac{e}{4}=1 \[2ex]

So,\[2ex]

\displaystyle a=1,\ b=\ -6,\ c=1,d=2,\ e=4\ \[2ex]

$