The fundamental theorem of algebra states that a polynomial of degree \( n \) will have exactly \( n \) roots (solutions), considering multiplicity and complex numbers.
a. The equation \(-9x + 4x^2 + 3 = 0\) is a quadratic equation (degree 2) and will have exactly 2 solutions.
b. The equation \(-2x^2 - 5 = 0\) is also a quadratic equation (degree 2), which means it also has 2 solutions.
c. The equation \(\frac{5}{7}x^2 = 0\) is quadratic with a degree of 2, hence it has 2 solutions, but both are the same (x = 0 has multiplicity 2).
d. The equation \(0x^2 + 7x - 2 = 0\) is linear (degree 1), so it has exactly 1 solution.
e. The equation \(0x^3 - x^2 + 2x + 4 = 0\) can be simplified to \(-x^2 + 2x + 4 = 0\) which is also quadratic (degree 2), thus it has 2 solutions.
In summary, the equations that have two solutions are a, b, and e.
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