<s>[INST] Irreducibility measure of integer polynomials (Partial) Question in short: Let $p$ be a monic integer polynomial of degree $n$. Is there a natural number $k$ with $0 \leq k \leq n$ such that $p+k$ is irreducible over the integers?
Longer version:
Let $p$ be a monic polynomial over the integers. Define the irreducibility measure $d(p)$ of $p$ as the smallest integer $k \geq 0$ such that $p+k$ is irreducible over the integers.
Define $M_n:=$ sup $\{ d(p) | deg(p)=n \}$ for $n \geq 2$. Here $deg(p)$ is the degree of $p$.

Question: Is it true that $M_n \leq n$? (Answer no, by Joachim König). Is there a good bound for $M_n$ ?

The question is based on some small computer experiments.
edit: Sorry I forgot the condition that the polynomials are monic (I did all computer experiments with that assumption. The answer by Joachim König gives a counterexample in the non-monic case).
 [/INST] For $f=x^6 - 3x^5 - 2x^4 + 10x^3 + x^2 - 8x - 5$, one has
$$f=(x^3 - 2x^2 - 2x + 5)(x^3 - x^2 - 2x - 1),$$
$$f+1=(x-2)(x^5 - x^4 - 4x^3 + 2x^2 + 5x + 2),$$
$$f+2=(x^2-x-1)(x^4 - 2x^3 - 3x^2 + 5x + 3),$$
$$f+3=(x^2-2)(x^4 - 3x^3 + 4x + 1),$$
$$f+4=(x+1)(x^5 - 4x^4 + 2x^3 + 8x^2 - 7x - 1),$$
$$f+5=x(x^5 - 3x^4 - 2x^3 + 10x^2 + x - 8),$$
$$f+6=(x-1)(x^5 - 2x^4 - 4x^3 + 6x^2 + 7x - 1).$$
(Found more or less by brute force.)
PS: In order to not cause unnecessary confusion, the "answer" giving a counterexample for the ``non-monic case", currently mentioned in the OP, was $f(x)=6x^2+7x$; this was removed as a separate answer after the OP had been altered.
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