How many solutions does this system of equations have?𝑦=𝑥2 + 𝑥 + 3𝑦=−2⁢𝑥 − 5 A. no real solutions B. 1 real solution C. 2 real solutions D. 3 real solutions

To find the number of solutions to the system of equations, we need to set the two equations equal to each other and solve for x.

The two equations are:

1. y = x^2 + x + 3
2. y = -2x - 5

Setting them equal to each other gives us:

x^2 + x + 3 = -2x - 5

Rearranging the terms, we get:

x^2 + 3x + 8 = 0

This is a quadratic equation, and we can find the number of solutions by calculating the discriminant (b^2 - 4ac). In this equation, a = 1, b = 3, and c = 8.

The discriminant is (3)^2 - 418 = 9 - 32 = -23

Since the discriminant is less than 0, the quadratic equation has no real solutions. Therefore, the system of equations has no real solutions.

So, the answer is A. No real solutions.