Is this ideal a prime ideal?

Question

Let $k$ be a field and $k[x_1,x_2,x_3,x_4]$ a polynomial ring in four variables over $k$. How can we show that the ideal $(x_3^3-x_2^2x_4, x_4^3-x_1^2x_3, x_3x_4-x_1x_2, x_2x_4^2-x_1x_3^2)$ is prime? I always find this kind of problem is not easy to me, is there a criterion for this kind of problem?