How to control this latch with Positive Logic

Generally, an RS latch will be used in situations where both inputs will never be active simultaneously, and thus the behavior of the circuit in such cases won't matter. The effect of the extra "or" gates is to make it so that when both inputs are true, both outputs will also be true; in their absence, having both inputs be true would force both outputs to be false. If one needs the outputs to both be true when both inputs are true, then the extra gates will be required. In cases where it would be acceptable for both outputs to be false as long as both inputs were true, however, the OR gates may be omitted (simply wire the outputs to the input that came from a NOR gate) since they do not affect behavior in other cases.

You may be looking for this:

CD4043/CD4044 3-state SR latch with common enable.

You might way to use the common enable in the CD4044 to implement the solution you're looking for. Connect all 4 R inputs to Vcc (HIGH), then use E as a common reset. For this to work you need a pull-down resistor on every output. Some MCUs inputs can be configured with internal pull-ups/downs, so you might be able to do this with no additional parts.

Most MCUs inputs can't be configured with internal pull-downs, only with pull-ups. So you may then want to consider another alternative. Use CD4043 instead of CD4044, then connect all 4 R inputs to GND (LOW) and use E as an active low common reset. You will then need pull-ups on every output instead of pull-downs, so just use the pull-ups of the MCU inputs by configuring it accordingly.

Why does this work?

If you look at the truth table of CD4044:

1. When E is HIGH it's equivalent to set all 4 R inputs to HIGH (= you get the same output from either E or R).
2. When E is LOW all the outputs will be in high-impedance (open circuit) and the pull-downs of the MCU will force a LOW on them regardless of the S inputs (= it effectively acts as a reset).

You can derive a similar deduction for CD4043.