= Solution
For a complex bundle $E$, write $c_t(E)=\sum_{j\geq0}c_j^{MU}(E)t^j$, with $c_0(E)=1$ and $c_j(E)=0$ for $j>\operatorname{rank}_{\mathbb C}E$. The <Whitney sum formula for Chern classes> in <complex cobordism> is
$$
\boxed{c_t(\eta\oplus\xi)=c_t(\eta)c_t(\xi),\qquad
c_k(\eta\oplus\xi)=\sum_{a+b=k}c_a(\eta)c_b(\xi).}
$$
Products are in the complex-cobordism cohomology ring; these classes have even degrees $2j$. The <splitting principle for complex vector bundles> explains the formula: on a splitting space the <Chern classes> are elementary symmetric functions of the first <Chern classes> of the line summands, and joining the two lists multiplies their total Chern polynomials.
Use the line convention for the <projective bundle>:
$$
\mathbb{CP}(\eta)=\{(x,\ell):x\in X,\ \ell\subset\eta_x\text{ is a complex line}\}.
$$
Its <relative tautological line bundle> is
$$
\eta(1)=\{((x,\ell),v):v\in\ell\}\longrightarrow\mathbb{CP}(\eta).
$$
It is a complex rank-one subbundle of $p^*\eta$. Choose a <Hermitian metric> and let $E=\eta(1)^\perp$, of rank $n-1$, so $p^*\eta=\eta(1)\oplus E$.
Throughout, $x=c_1^{MU}(\eta(1))$, the tautological line's class, as specified before the final equation. Put $a_k=p^*c_k^{MU}(\eta)$ and $b_k=c_k^{MU}(E)$, with $a_0=b_0=1$. Since $c_t(\eta(1))=1+xt$, Whitney multiplication gives
$$
\sum_{k=0}^{n}a_kt^k=(1+xt)\sum_{k=0}^{n-1}b_kt^k.
$$
Comparison of coefficients gives $a_k=b_k+xb_{k-1}$, hence recursively $b_k=a_k-xb_{k-1}$. Therefore the <Chern classes of a tautological-line complement> are
$$
\boxed{c_k^{MU}(\eta(1)^\perp)
=\sum_{j=0}^{k}(-1)^j x^j\,p^*c_{k-j}^{MU}(\eta)
\quad(0\leq k\leq n-1),}
$$
and the classes for $k\geq n$ vanish by the rank.
At degree $n$ the same recursion has $b_n=0$, giving
$$
0=a_n-xa_{n-1}+x^2a_{n-2}-\cdots+(-1)^nx^n.
$$
Multiplying by $(-1)^n$ proves the <projective bundle relation in complex cobordism>
$$
\boxed{x^n-p^*c_1^{MU}(\eta)x^{n-1}
+p^*c_2^{MU}(\eta)x^{n-2}-\cdots+(-1)^np^*c_n^{MU}(\eta)=0.}
$$
These alternating signs come from the inverse formal power series $(1+xt)^{-1}$. They do not require identifying the <First Chern class> of a dual line with $-x$, which is generally incorrect in <complex cobordism>.
The final sentence's designation $x=c_1(\eta)$ must mean the earlier $c_1(\eta(1))$. If instead one substitutes $p^*c_1(\eta)$ literally, the claimed relation is false even in ordinary cohomology. For example, take $\eta=L\oplus L$ over $\mathbb{CP}^2$, with $a=c_1(L)$ a generator. Its <projective bundle> is $\mathbb{CP}^2\times\mathbb{CP}^1$, and $p^*a^2\ne0$. With the incorrect substitution $x=2p^*a$, the rank-two polynomial evaluates to
$$
(2p^*a)^2-(2p^*a)(2p^*a)+p^*a^2=p^*a^2\ne0.
$$
The tautological-line interpretation above is the one consistent with the bundle decomposition and the stated relation.
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