I think you were asking how many roots does the polynomial 7+5r⁴-32² have in all
For no. of total roots of a polynomial, we notice the highest power of the variable. Here, the variable is 4, whose highest power (degree) is 4. So, the polynomial above will have 4 roots in all.
And if you're asked to calculate no. of real roots or no. of complex roots, then Rather than using The concept of derivates, you can use Descarte's Sign Rule
Let, we call the polynomial, f(r) = 7+5r⁴ - 32²
We need to write the polynomial in simplest form, if we do, we will be having;
→ f(r) = 5r⁴ - 1017
Note the sign change of each term, how many times the sign changed , like from +ve to +ve doesn't counts, +ve to -ve counts 1 and then -ve to +ve again counts 1, so no. of sign change of f(r) will give the total no. of +ve real roots
Now, replace r by -r, and do the same, notice no. of sign changes, so no. of sign change of f(-r) will give the total no. of -ve real roots
So, we will be having total no. of real roots, as the sum of those obtained in above two cases.
And, no. of complex roots will be given by Degree (highest power) - no. of real roots,as the root can only be of two natures either real or complex. If you apply Descarte's Rule in the above polynomial, you will obtain two complex and two real roots (Refer for attachment for better understanding)