Suppose $f(x)=x^5+12x^4+21x^2-30$. Prove that $f(x)$ is an irreducible element of $\mathbb{Z}[x]$.
Ideas: We can try and prove this using a contradiction. Suppose that $f(x)=g(x)h(x)$, then we have $deg(f)=deg(g)+deg(h)$ and so $5=deg(g)+deg(h)$. Therefore the possible degrees of $g,h$ are $1,2,3,4$. I know that for a polynomial of degree $2$ or $3$ we have that the polynomial is irreducible if and only if there is no root in the field which we are working in, but that doesn't seem applicable here.
I don't want to assume eisenstein's criterion or any other machinery than the basics.
Since $f$ has no rational root by the rational root test, we can assume that it is a product of a quadratic and a cubic polynomial. Over the field $\Bbb F_7$, $f$ is irreducible, because we can check that $$ f=(x^2+ax+b)(x^3+cx^2+dx+e) $$ is impossible over $\Bbb F_7$ by very elementary equation solving (comparison of coefficients gives equations in $a,b,c,d,e\in \Bbb F_7$). Hence $f$ is irreducible over $\Bbb Z$ as well. If you don't want this modular reduction as well, then you still can show directly that the Diophantine equations over $\Bbb Z$ have no solution.
On the other hand, this polynomial is made for Eisenstein. Perhaps it is a motivation to study Eisenstein - which is sort of basic in this area.