Factorise 12n2+n−165 completely.

Sure, here are the steps to factorise the expression 12n^2 + n - 165:

Step 1: Multiply the coefficient of n^2 (which is 12) with the constant term (which is -165). The result is -1980.

Step 2: Look for two numbers that multiply to -1980 (from step 1) and add up to 1 (the coefficient of n). The numbers are 45 and -44.

Step 3: Rewrite the middle term of the original expression as the sum of the terms 45n and -44n. This gives us 12n^2 + 45n - 44n - 165.

Step 4: Factor by grouping. The first two terms can be grouped together, and the last two terms can be grouped together. This gives us 3n(4n + 15) - 11(4n + 15).

Step 5: Notice that (4n + 15) is a common factor. Factor out this term to get (4n + 15)(3n - 11).

So, the expression 12n^2 + n - 165 factorises to (4n + 15)(3n - 11).