= Solution
On a <complex manifold>, a <divisor on a complex manifold> is a locally finite formal sum $D=\sum_Y m_Y Y$, where $Y$ ranges over irreducible <complex analytic hypersurfaces> and $m_Y\in\mathbb Z$. The local algebra needed here is that the <local ring> $\mathcal O_{X,x}$, identified in coordinates with convergent power-series germs, is a <regular local ring> and a <unique factorization domain>. In particular, each height-one prime is generated by one irreducible germ, and a nonzero meromorphic germ is a quotient of holomorphic germs. Reduced local equations of the same hypersurface differ by a holomorphic unit. These facts include local branches when a globally irreducible hypersurface has several branches at a singular point.
The <order of a meromorphic function along a hypersurface> is the exponent of its reduced local equation in the quotient's factorization. Equivalently, at a generic smooth point of $Y$, write $f=t^m u$, with $t$ a transverse local equation of $Y$ and $u$ a holomorphic unit; then $\operatorname{ord}_Y(f)=m$. Changing a reduced equation multiplies it by a unit and leaves $m$ unchanged. These local orders agree on overlaps and are constant along the generic smooth locus of an irreducible hypersurface. A nonzero global <meromorphic function> therefore defines its <principal divisor on a complex manifold>
$$
\operatorname{div}(f)=\sum_Y\operatorname{ord}_Y(f)Y.
$$
Positive coefficients record zeros and negative coefficients record poles.
A local defining function $f_i$ for $D$ on $U_i$ is a nonzero <meromorphic function> whose orders there equal the coefficients of $D$. It can be constructed as the product of local reduced hypersurface equations to the prescribed integer powers. On an overlap, $f_i/f_j$ has no zero or pole along any hypersurface. The <unique factorization domain> property then makes it a nowhere-zero <holomorphic function>, after choosing representatives of the germs. The <holomorphic line bundle associated to a divisor> is obtained by gluing holomorphic frames using
$$
e_i=\frac{f_j}{f_i}e_j.
$$
These ratios satisfy the cocycle condition, and the expressions $f_i e_i$ agree on overlaps. They form the canonical <meromorphic section of a holomorphic line bundle> with divisor $D$.
If $D=\operatorname{div}(f)$ globally, $f_i/f$ is a holomorphic unit on $U_i$, and $(f_i/f)e_i$ is a global nowhere-zero <holomorphic section>. Hence $[D]$ is holomorphically trivial. Conversely, if $[D]$ has a nowhere-zero global holomorphic frame $e$, write $e_i=a_i e$ with $a_i$ holomorphic units. The coefficients $f_i a_i$ of the canonical <meromorphic section of a holomorphic line bundle> agree on overlaps and give a nonzero global <meromorphic function> $f$. Since multiplication by a unit does not change orders, $\operatorname{div}(f)=D$. Therefore
$$
\boxed{[D]\cong\mathcal O_X\quad\Longleftrightarrow\quad D\text{ is principal}.}
$$
For a smooth hypersurface $Y\subset X$, the <adjunction formula> is
$$
\boxed{K_Y\cong\bigl(K_X\otimes[Y]\bigr)|_Y.}
$$
Indeed, the <holomorphic conormal sequence> has first term $N_{Y/X}^*\cong[-Y]|_Y$; taking its determinant yields $K_X|_Y\cong N_{Y/X}^*\otimes K_Y$, which gives the formula. Here the <canonical bundle> is the top exterior power of the holomorphic cotangent bundle.
Apply this first to $W\subset\mathbb P^4$. Its degree-$d_1$ equation is a section of $\mathcal O(d_1)$, so $[W]\cong\mathcal O(d_1)$. Since $K_{\mathbb P^4}\cong\mathcal O(-5)$, we obtain $K_W\cong\mathcal O_W(d_1-5)$. The restricted second equation cuts out $S$ in $W$ and has associated bundle $\mathcal O_W(d_2)$. A second application of <adjunction formula> gives the <canonical bundle of a regular projective complete intersection>
$$
K_S\cong\mathcal O_S(d_1+d_2-5).
$$
It remains to prove that a nonzero exponent cannot yield a trivial <holomorphic line bundle> on $S$. Choose a connected component $S_0$ and a hyperplane meeting but not containing it. Such a hyperplane exists: a positive-dimensional compact projective submanifold cannot lie entirely in an affine chart, since all its affine coordinate functions would be global <holomorphic functions> and hence constant. A hyperplane through one point and not another consequently meets $S_0$ properly. Its equation gives a <holomorphic section> of $\mathcal O_{S_0}(1)$ that has both a zero and a nonzero value. For $m>0$, the $m$th power has the same two properties. Were $\mathcal O_{S_0}(m)$ trivial, this section would be represented by a global <holomorphic function>, contradicting its constancy on compact connected $S_0$. For $m<0$, triviality would imply triviality of its positive dual power. This proves <nontrivial hyperplane powers on a compact projective submanifold> and hence
$$
\boxed{d_1+d_2=5,\qquad
(d_1,d_2)=(1,4),(2,3),(3,2),(4,1).}
$$
The degrees are positive because the two equations define a hypersurface and a surface of the specified dimensions.
The calculation uses “non-singular” in the usual smooth complete-intersection sense: the equations define reduced smooth divisors, with independent differentials along $S$. If only their reduced zero sets are required to be smooth, polynomial multiplicities cannot determine the <canonical bundle>. For example, $Q_1=Z_0^m$ and $Q_2=Z_1^4+Z_2^4+Z_3^4+Z_4^4$ have the same smooth quartic surface in the hyperplane $Z_0=0$ for every $m\geq1$. That surface has trivial <canonical bundle>, while the displayed polynomial degree sum is $m+4$. Thus unreduced equations must not be substituted into the <adjunction formula> for the reduced surface.
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