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We consider an equation of the type $$a_1^{}{x}_1^2+\cdots +a_n^{}{x}_n^2=b{x}_1\cdots x_n$$ over the finite field ${𝔽}_q={𝔽}_{p^s}$. Carlitz obtained formulas for the number of solutions to this equation when $n=3$ and when $n=4$ and $q\equiv 3(mod4)$. In our earlier papers, we found formulas for the number of solutions when $d=gcd(n-2,(q-1)/2)=1$ or $2$ or $4$; and when $d\>1$ and $-1$ is a power of $p$ modulo $2d$. In this paper, we obtain formulas for the number of solutions when $d=2^t$, $t\ge 3$, $p\equiv 3\text{or}5(mod8)$ or $p\equiv 9(mod16)$. For general case, we derive lower bounds for the number of solutions.