About the correction for this question- is the first TM suppose to stay a single-tape TM and the second should be a two-tape TM?

And another question- I can show that if there is a TM that halts on every input x in $2010|x|^2+1$ steps, there is a TM that halts on every input x in about $|x|^2+|x|$ steps.

Is the requirement for $|x|^2+1$ steps is strict, or $|x|^2+|x|$ is good enough? (I can't avoid starting with going over the entire tape before I can even start reducing the TM M number of steps, and therefore I have an extra |x|)