I don't know if you are more familiar with matrix notation

  phi(x) = [x1, x2, x1² + x2²]
  phi(x)· phi(y) = [x1, x2, x1² + x2²]*[y1, y2, y1² + y2²]
  phi(x)· phi(y) = x1 y1 + x2 y2 + (x1² + x2²)(y1² + y²)