Answer:
The conic section basis of the equation x^2 + xy + y^2 + 2x = -3 can be determined by examining the coefficients of the x^2, xy, and y^2 terms.
To do this, we can start by completing the square for the quadratic terms in the equation:
x^2 + xy + y^2 + 2x = -3
(x^2 + 2x) + xy + y^2 = -3
(x + 1)^2 - 1 + xy + y^2 = -3
(x + 1)^2 + xy + y^2 = -2
Now, we can see that the coefficient of the xy term is positive, which indicates that the conic section is an ellipse. Specifically, this is a rotated ellipse because the x and y terms are not squared separately and have a non-zero coefficient.
Therefore, the conic section basis of the equation x^2 + xy + y^2 + 2x = -3 is a rotated ellipse.
Explanation: