Show that Discriminant $D$ of
$x^4+7x^3+14x^2+3x-9$
is $0$ WITHOUT using the discriminant formula. If this is true, what root properties does this specific polynomial hold and do the factors (for a certain $x$ value evaluated at this polynomial) have a specific form?
If the discriminant is zero, there are multiple roots (have a look here).
By inspection $x=-3$ is a root of equation $x^4+7x^3+14x^2+3x-9=0$. Using long division $$x^4+7x^3+14x^2+3x-9=(x+3)(x^3+4 x^2+2 x-3)$$ By inspection $x=-3$ is a root of equation $x^3+4 x^2+2 x-3=0$. Continuing with long division$$x^4+7x^3+14x^2+3x-9=(x+3)^2(x^2+x-1)$$