We can factor this polynomial using grouping. What we do is split the 36p into into parts so that we can factor the polynomial one half at a time.
To do so, we first multiply the leading coeffient by the constant. In other words, a times c where [tex]ap^2+bp+c[/tex]. After multiplying 4 and 81, we get 324. Now we need to find two numbers that multiply to be 324 and add to be 36.
The factors of 324 are: 1, 2, 3, 4, 6, 9, 12,18, 27, 36, 54, 81, 108, 162, and 324.
The pair of numbers that works in this case are 18 and 18 since 324 is a perfect square.
Now we split the 36p:
[tex](4p^2+18p)+(18p+81)[/tex]
Now factor both halves.
[tex]2p(2p+9)+9(2p+9)[/tex]
We have a common binomial of 2p+9 that can be factored out it get
[tex](2p+9)(2p+9)[/tex]
or
[tex](2p+9)^2[/tex]
