Please excuse me if the question seems to commonplace. Can Mathematica give a response to this question:

In which condition related to the coefficients $c>0, \, n\geq 2, 0<p<1,\, a>c, \, \, \text{and} \, \, b>0$, this equation

$$ F(x)= -c x^{n+p} -bx^n + (a-c)x^p -b \qquad (*) $$

have solution in $\mathbb R$.

Assumptions:

assum = And @@ {c > 0, n >= 2, 0 < p < 1, a > c, b > 0}

Function:

f[x_] := -c x^(n + p) - b x^n + (a - c) x^p - b

And I find this way to do it

assum = And @@ {c > 0, n >= 2, 0 < p < 1, a > c, b > 0}
f[x_] := -c x^(n + p) - b x^n + (a - c) x^p - b
Assuming[assum, FullSimplify@Reduce[f[x] == 0, Reals]]