$n$ 是正整数。 证明：对于所有的 $n$，$6n^2 + 25n + 11$ 都不是质数。 由此，判断 16261 是否为质数。清晰地写出你的解答过程。

1. Factorise the quadratic expression.

$6n^2+25n+11=(2n+1)(3n+11)$

2. Use the factorisation to justify that the expression is not prime.

Since $n$ is a positive integer, $2n+1$ and $3n+11$ are integers.

Also, $2n+1>1$ and $3n+11>1$.

Therefore $6n^2+25n+11$ is a product of two integers greater than 1, so it is not a prime number.

1. Equate the given number to the quadratic expression from part (a) and solve for $n$.

Using part (a), consider
$6n^2+25n+11=16261.$
Then
$6n^2+25n-16250=0.$
Solving gives
$n=50$
since
$6(50)^2+25(50)+11=15000+1250+11=16261.$

2. Use the factorisation from part (a) to conclude whether 16261 is prime.

From part (a),
$6n^2+25n+11=(2n+1)(3n+11).$
When $n=50$,
$16261=(2(50)+1)(3(50)+11)=101\times161.$
Since $101>1$ and $161>1$, $16261$ has factors other than 1 and itself.