Number of real roots for $x^4+5|x^3|+3x^2+20|x|+10=0$ are?

Number of real roots for $x^4+5|x^3|+3x^2+20|x|+10=0$ are ?

Approach:-

I used Descarte's rule of signs to get an idea of how many real roots does a polynomial equation have, so I thought of making 2 cases for x>0 and x<0

Case 1:- when x>0

f(x)=$x^4+5x^3+3x^2+20x+10=0$

now there are 0 sign changes here so that means 0 positive real roots

now I will try for f(-x) = $x^4-5x^3+3x^2-20x+10=0$

here we have 4 sign changes indicating at most there can be 4 negative real roots, so I tabulated the possibilities which we can get

1. 4 negative real roots, 0 imaginary roots
2. 2 negative real roots , 2 imaginary roots
3. 4 imaginary roots

Now I did same procedure for Case 2:- when x<0

f(x)=$x^4-5x^3+3x^2-20x+10=0$

here we have 4 sign changes which means we can have at most 4 positive real roots

f(-x)=$x^4+5x^3+3x^2+20x+10=0$

here 0 sign changes so no negative real roots possible

so the possibilities from this case would be:-

1. 4 positive real roots, 0 imaginary roots
2. 2 positive real roots , 2 imaginary roots
3. 4 imaginary roots

how to proceed from here (using this method) ?

Solution 1:

There is no real root of this equation since the function on left have only nonnegative terms and the last one is strictly positive, so their sum can never be $0$ (for real $x$).