To simplify this division of the two fractions, we first need to find a common denominator for the two fractions.

The common denominator for the two fractions is the product of their denominators, which is (16m²-1)(16m²-8m+1).

Now, we multiply each fraction by the appropriate form of one in terms of the other fraction's denominator:

[(M²+5M)/(16m²-1)] [(16m²-8m+1)/(16m²-8m+1)] / [(m⁴+125m)/(16m²-8m+1)] [(16m²-1)/(16m²-1)]

This gives us:

[(M²+5M)(16m²-8m+1)] / [(16m²-1)(m⁴+125m)]

Expanding out the numerators and denominators gives:

(16M^2^3-6M+M^2-5M) / (16m^6+200m^2-16m^4-200m)

Now we combine like terms in the numerator and denominator to simplify further:

(16M^4-M-5M) / (16m^6-16m^4+200m^2-200m)

Finally, the simplified form of the division of the two fractions is:

(16M^4-6M) / (16m^6-16m^4+200m^2-200m)