Factorise 27x2+42x+11 completely.

The given expression is 27x^2 + 42x + 11.

First, we need to multiply the coefficient of x^2 (which is 27) and the constant term (which is 11). So, 27*11 = 297.

Now, we need to find two numbers that multiply to 297 and add to 42 (the coefficient of x). The numbers 9 and 33 fit these criteria because 9*33 = 297 and 9 + 33 = 42.

Next, we rewrite the middle term of the expression (42x) as the sum of 9x and 33x. This gives us 27x^2 + 9x + 33x + 11.

Now, we can factor by grouping. The greatest common factor of the first two terms is 9x, and the greatest common factor of the last two terms is 11. This gives us 9x(3x + 1) + 11(3x + 1).

Finally, we notice that (3x + 1) is a common factor, so we can factor it out to get (3x + 1)(9x + 11).

So, the expression 27x^2 + 42x + 11 factorises to (3x + 1)(9x + 11).