Probability that quadratic is an ellipse Let $p(x,y)$ be a quadratic equation with terms
$$ x^2 + y^2 + xy + x + y + 1\;, $$
where each term is multiplied by a random real coefficient uniformly 
distributed in $[-1,1]$.

Q. What is the probability that $p(x,y)=0$ is an ellipse?


          


          

Random quadratics.


Perhaps it is about 25%?

In terms of $A_Q$, the $3\times 3$ matrix of the quadratic equation
$$Q(x,y) = Ax^2+Bxy+Cy^2+Dx+Ey+F=0,$$ 
i.e., 
$$A_Q = 
\begin{pmatrix}
  A & B/2 & D/2 \\
  B/2 & C & E/2 \\
  D/2 & E/2 & F
\end{pmatrix},$$
the equation represents a real nondegenerate ellipse iff 


*

*$\det(A_{33}) > 0$, and 

*$(A+C)\det(A_Q) < 0$


where $A_{33}$ is the $2\times 2$ upper-left submatrix of $A_Q$:
$$A_{33} = \left( \begin{matrix}A & B/2\\B/2 & C\end{matrix}\right).$$
NB: Both conditions (1) and (2) are homogeneous; i.e., for any $k>0$, they hold for $(A,B,C,D,E,F)$ iff they hold for $(kA,kB,kC,kD,kE,kF)$. Consequently, the probability that the i.i.d. coefficients correspond to a real nondegenerate ellipse is invariant under rescaling of the distribution; e.g., if they are i.i.d. $\text{Uniform[-k,k]}$, then the answer is the same for all $k>0$.    

Simulations
R code
Pending analytical solution of the posted question, here's a simulation (in R code, which defaults to using a Mersenne Twister PRNG) that simply samples the six coefficients (from $\text{Uniform}[-1,1]$) repeatedly and finds the proportion of times both of the above conditions are met:
library(conics)
nsamp = 10^8
count = 0
for (i in 1:nsamp){
  coeffs = runif(6,-1,1)      # sample the six coefficients
  AQ = conicMatrix(coeffs)    # AQ is the matrix of the conic
  det.AQ = det(AQ)
  det.A33 = det(AQ[1:2,1:2])  # det.A33 is -discrimant/4
  if (det.A33 > 0 & (coeffs[1]+coeffs[3])*det.AQ < 0){  
    count = count + 1
  }
}
print(count/nsamp)

Output:
 0.2738687

An approximate 95% confidence interval is therefore $0.2739 \pm 0.0001$.
C code
Essentially the same algorithm programmed in C, using two different PRNGs, gave the following consistent results for a much larger sample size:
PRNG                   Sample Size   95% Confidence Interval
------------------------------------------------------------
(a) Mersenne Twister      10^11        0.273898 +- 0.000003
(b) PCG                   10^11        0.273896 +- 0.000003
------------------------------------------------------------
(combined)              2 10^11        0.273897 +- 0.000002

(a) mt19937-64, (b) pcg-c-basic-0.9.
