Alan

Do it as follows:

See https://web2.0calc.com/questions/pre-calc-problem#r5 

Here's one way of showing it:

As follows:

Can you finish it from here?

As follows:

2n^2 + 3n + 11 - n^2 + 9n  can be written as n^2 + 12n + 11 which, in turn can be factored as (n + 1)(n + 11).

So what does this tell you, remembering that a prime number has no factors other than itself and 1?

If x is a positive even integer then so is 9x -2, so the smallest odd number greater than this must be 9x-2+1 = 9x-1

Similarly, 8x+3 must be an odd number, so the largets odd number less than this must be 8x+3-2 = 8x + 1

Can you finish from here?

For 

a) How many ways are there of choosing 4 from 14

b) How many ways of choosing 4 from 14 less how many ways of choosing 2 from 12

c) How many ways of choosing 4 from 12 plus how many ways of choosing 2 from 12.

(In general:  How many ways of choosing m from n is given by:  \(\frac{n!}{(n-m)!\times m!}\) )

sin^2(x)+sin(x).cos^2(x) does NOT equal tan(x) as a quick plot shows:

Here's an alternative approach: