First, solve the linear equation \(2x + y + 4 = 0\) for \(y\):
\(y = -2x - 4\).
Substitute \(y = -2x - 4\) into the curve equation \(2xy + 5y^2 = 24\):
\(2x(-2x - 4) + 5(-2x - 4)^2 = 24\).
Simplify and expand:
\(-4x^2 - 8x + 5(4x^2 + 16x + 16) = 24\).
\(-4x^2 - 8x + 20x^2 + 80x + 80 = 24\).
Combine like terms:
\(16x^2 + 72x + 56 = 0\).
Divide the entire equation by 8:
\(2x^2 + 9x + 7 = 0\).
Factor the quadratic equation:
\((2x + 7)(x + 1) = 0\).
Set each factor to zero and solve for \(x\):
\(2x + 7 = 0 \Rightarrow x = -\frac{7}{2}\).
\(x + 1 = 0 \Rightarrow x = -1\).
Substitute \(x = -\frac{7}{2}\) into \(y = -2x - 4\):
\(y = -2\left(-\frac{7}{2}\right) - 4 = 3\).
Substitute \(x = -1\) into \(y = -2x - 4\):
\(y = -2(-1) - 4 = -2\).
Thus, the points of intersection are \((-1, -2)\) and \(\left(-\frac{7}{2}, 3\right)\).
