factor 4x^{2}+36x+81

The given expression is a trinomial, which can be factored into two binomial terms.

To factor 4x^2 + 36x + 81, we first look for two numbers that multiply to 4*81=324 and add to 36.
These numbers are 18 and 18.

Therefore, we can rewrite the expression as:
4x^2 + 36x + 81 = 4x^2 + 18x + 18x + 81

Now, we can factor by grouping:
= 4x(x + 9) + 18(x + 9)
= (4x + 18)(x + 9)
= 2(2x + 9)(x + 9)

Therefore, the factored form of 4x^2 + 36x + 81 is 2(2x + 9)(x + 9).