If $\alpha$ and $\beta$ are roots of the quadratic equation $2p^{2}x^{2}+2p^{3}x-1=0, p \in R-{0}$, then minimum value of $\alpha^{4}+\beta^{4}$ is

For 2p^2x^2 + 2p^3x - 1 = 0, the sum of roots alpha + beta = -p and product alpha*beta = -1/(2p^2). We want to minimize alpha^4 + beta^4 = (alpha^2 + beta^2)^2 - 2(alpha*beta)^2. Substituting values, we get ((-p)^2 - 2(-1/(2p^2)))^2 - 2(-1/(2p^2))^2 = (p^2 + 1/p^2)^2 - 1/(2p^4). Minimizing this expression leads to 2 + sqrt(2).