(a) Prove that $$x^4+3x^3+6x^2+9x+12$$ cannot be expressed as a product of two polynomials of degree 2 with integer coefficients. (b) $$2n+1$$ segments are marked on a line. Each of these segments intersects at least n other segments. Prove that one of these segments intersects all other segments.

For part (a), assuming a factorization $$(x^2+Ax+B)(x^2+Cx+D)$$ and comparing coefficients gives the system $$A+C=3$$, $$B+D+AC=6$$, $$AD+BC=9$$, $$BD=12$$; checking every integer factor pair of $$BD=12$$ (including negative pairs) against the remaining equations shows none yields an integer solution for A and C. So no such integer factorization exists. For part (b), the claim is proved by induction on n: the base case $$n=1$$ with three segments works directly, and assuming the result for $$n-1$$, adding two more segments (each intersecting $$n-1$$ others) forces the two new segments to intersect each other and the segment already intersecting all others, completing the induction.