Factoring the second term shows you what the common denominator must be. Then multiply the first term by the appropriate version of 1 and add.
[tex]\displaystyle\frac{4}{m+9}+\frac{5}{m^2-81}=\frac{4}{m+9}+\frac{5}{(m+9)(m-9)}\\\\=\frac{4(m-9)}{(m+9)(m-9)}+\frac{5}{(m+9)(m-9)}=\frac{4(m-9)+5}{(m+9)(m-9)}\\\\=\frac{4m-36+5}{(m+9)(m-9)}\\\\=\frac{4m-31}{(m+9)(m-9)}[/tex]
