Find all real numbers $k$ such that $x^4+kx^3+x^2+4kx+16=0$ is true for exactly one real number $x = r$.  Enter all the possible values of $k,$ separated by commas.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Find all real numbers $k$ such that $x^4+kx^3+x^2+4kx+16=0$ is true for exactly one real number $x = r$.  Enter all the possible values of $k,$ separated by commas.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\frac{9}{4}, -\frac{9}{4}
Find all real numbers $k$ such that $x^4+kx^3+x^2+4kx+16=0$ is true for exactly one real number $x = r$.  Enter all the possible values of $k,$ separated by commas.
train