Plotting bode plot of the transfer function
First, cancel out the pole and zero:

\$H(s) = 250s/(s+25)\$

Next, determine the pole location:

\$H(s) = 10s/(1+s/25)\$

The break frequency is 25 rad/s.

Since there is a zero at origin, the slope until 25 rad/s is +20 dB and the slope after 25 rad/s is 0 dB.

We can start the graph at 1/10th of the lowest break frequency: 2.5 rad/s but that's quite an odd number.  Let's select 1 rad/s instead.

The gain at 1 rad/s is:

\$H(1) = 10*1/(1)\ = 10 = 20 \textbf{dB}\$

This does not directly tell us the gain at and beyond the 25 rad/s break frequency.  We know that the gain difference between the gains at the break frequency and at 1/10th of the break frequency is 20 dB.

Thus, let's determine the gain at 2.5 rad/s:

\$H(2.5) = 10*2.5/(1)\ = 25 = 28 \textbf{dB}\$

Gain at 1/10th of the break frequency is 28 dB.  One can draw vertical line from 20 dB at 1 rad/s through 28 dB at 2.5 rad/s to 48 dB at 25 rad/s.  At 25 rad/s, the gain slope becomes zero and hence one can draw a horizontal line at 48 dB to ~1,000 rad/s.

Also, notice that the DC gain is 0, which is a number that cannot be represented on a logarithmic scale.

\$H(0) = 10*0/(1)\ = 0\$

MatLab: