Can the polynomial equation have exactly 1 real solution, including any repeated solutions? (Use the Fundamental Theorem of Algebra and the Complex Conjugates Theorem.) - 9x3+19x2+17=0 yes or no

yes, there are infinitety many polynomial that have exactly one real root just like your example, to determine the real root first let the real root is a, and the complex roots are b±ic the polynomial satisfy

-9x³ + 19x² + 17 = - (x - a) (x - b - ic) (x - b + ic)

9x³ - 19x² - 17 = (x - a) (x - b - ic) (x - b + ic)