To use the elimination method, we can set the two equations equal to each other:
x^2 + 2x - 4 = -x - 6
Now, let's rearrange the equation to set it equal to zero:
x^2 + 3x - 2 = 0
Now, we have a quadratic equation that we can solve using quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
x = (-3 ± sqrt(3^2 - 4*1*(-2))) / 2*1
x = (-3 ± sqrt(9 + 8)) / 2
x = (-3 ± sqrt(17)) / 2
Therefore, the solutions for x are:
x = (-3 + sqrt(17)) / 2
x = (-3 - sqrt(17)) / 2
Now, we can plug these values of x back into one of the original equations to find the corresponding values of y:
For x = (-3 + sqrt(17)) / 2:
y = (-3 + sqrt(17))^2 + 2(-3 + sqrt(17)) - 4
y = (9 - 6sqrt(17) + 17) - 6 + 2sqrt(17) - 4
y = 16 - 4
y = 12
For x = (-3 - sqrt(17)) / 2:
y = (-3 - sqrt(17))^2 + 2(-3 - sqrt(17)) - 4
y = (9 + 6sqrt(17) + 17) - 6 - 2sqrt(17) - 4
y = 26 - 4
y = 22
Therefore, the solutions to the system of equations are:
x = (-3 + sqrt(17)) / 2, y = 12
x = (-3 - sqrt(17)) / 2, y = 22