1. The statement "All quartic polynomial equations have at least one real solution" is false. Explanation: Quartic polynomials are degree 4 polynomials. According to the Fundamental Theorem of Algebra, a quartic polynomial has exactly 4 complex roots (counting multiplicities), but these roots may be real or complex. It is possible for a quartic polynomial to have no real roots if all roots are complex. 2. The statement "An infinite number of parabolas can share the same vertex, forming a family of quadratic functions" is true. Explanation: A parabola is defined by a quadratic function. If the vertex is fixed at a point $(h,k)$, the family of parabolas can be written as $y = a(x - h)^2 + k$ where $a$ is any real number. Since $a$ can take infinitely many values, infinitely many parabolas share the same vertex. 3. The statement "A polynomial inequality will either have no solution or infinitely many solutions" is false. Explanation: Polynomial inequalities can have finite solution sets or intervals. For example, $x^2 - 1 > 0$ has solutions $x < -1$ or $x > 1$, which is infinitely many solutions, but $x^2 + 1 < 0$ has no real solutions. However, some polynomial inequalities can have solutions consisting of finite intervals or unions of intervals, not just no or infinite solutions. Final answers: - Question 5: False - Question 6: True - Question 7: False