= Solution
Here the usual <homological algebra> convention is that the right-exact <functor> $F$ is additive. Module categories have enough <projective modules>, because any <module> is a quotient of a <free module>. Choose a <projective resolution>
$$
\cdots\xrightarrow{d_3}P_2\xrightarrow{d_2}P_1\xrightarrow{d_1}P_0\longrightarrow M\longrightarrow0.
$$
Apply $F$ to the unaugmented <chain complex> and define the <left derived functors> by
$$
\boxed{L_iF(M)=H_i(F(P_\bullet))=\frac{\ker F(d_i)}{\operatorname{im}F(d_{i+1})}\quad(i>0),\qquad L_0F(M)=\operatorname{coker}F(d_1).}
$$
Additivity ensures that $F$ preserves zero maps, sums of maps and homotopy identities, so the transformed differentials still compose to zero.
For a <module homomorphism> $u:M\to N$, lift it to a <chain map> between chosen <projective resolutions>. In degree zero, lift through $Q_0\twoheadrightarrow N$ using projectivity of $P_0$. In each higher degree, the already constructed map lands in the next kernel; exactness of $Q_\bullet$ and projectivity of $P_i$ supply the next lift. Any two lifts are <chain homotopic>: their degree-zero difference lands in $\ker(Q_0\to N)=\operatorname{im}d_1^Q$, so it lifts to $h_0:P_0\to Q_1$. Inductively lift the remaining difference after subtracting the preceding homotopy term. This gives
$$
u_i-v_i=d_{i+1}^Qh_i+h_{i-1}d_i^P,\qquad h_{-1}=0.
$$
Applying $F$ preserves this identity, so the lifts induce the same <homology (mathematics)> map. Identity lifts and composable lifts show that $L_iF$ is a <functor>. Comparison lifts between two resolutions over the identity of $M$ are inverse up to <chain homotopy>; hence the construction is independent of the chosen resolution up to its canonical natural <isomorphism>.
Right exactness identifies $L_0F(M)=\operatorname{coker}(F(P_1)\to F(P_0))$ naturally with $F(M)$. For a <projective module> $P$, the length-zero resolution gives $L_iF(P)=0$ for $i>0$. For a <short exact sequence> $0\to M'\to M\to M''\to0$, the <horseshoe lemma> builds a degreewise split short exact sequence of <projective resolutions>. Concretely, lift the projective generators covering $M''$ into $M$ and combine them with generators covering $M'$; repeat on the kernels. The middle projective term in each degree is the direct sum of the two outer ones. An <additive functor> preserves these split sequences, and lifting cycles, taking their differentials and returning to the subcomplex produces the connecting maps in
$$
\cdots\to L_2F(M'')\to L_1F(M')\to L_1F(M)\to L_1F(M'')\to F(M')\to F(M)\to F(M'')\to0.
$$
This is the natural long exact derived-functor sequence. In particular, if $0\to K\to P\to M\to0$ has $P$ projective, then $L_iF(M)\cong L_{i-1}F(K)$ for $i\ge2$, and $L_1F(M)=\ker(F(K)\to F(P))$. These are dimension-shifting formulas. Positive degrees can be killed by the surjection $P\to M$. If $F$ is exact, it preserves every resolution's exactness and all its positive <left derived functors> vanish. Conversely, if $L_1F$ vanishes on all modules, the displayed long exact sequence makes $F$ exact. A natural transformation between additive right-exact <functors> induces transformations on the <left derived functors> by applying it termwise.
For <extension of scalars> $F(M)=B\otimes_AM$, these derived functors are the <Tor functors> $\operatorname{Tor}_i^A(B,M)$. When $B=A/(a)$ and $a$ is a <non-zero-divisor>, there is the length-one <free resolution>
$$
0\longrightarrow A\xrightarrow{\ a\ }A\longrightarrow B\longrightarrow0.
$$
Although the construction above resolves $M$, one may instead use this resolution of $B$. To justify that calculation, tensor a free resolution of $M$ with the two-term resolution of $B$ to form a double complex. Taking homology first along the resolution of $B$ leaves $B\otimes_AP_\bullet$; taking it first along the resolution of $M$ leaves $M\xrightarrow{a}M$. The free terms preserve exactness under tensoring. The double complex is first-quadrant with finitely many terms on each total diagonal, so both filtrations compute the same total homology. This proves the required balancing directly.
The kernel and cokernel of multiplication by $a$ now give <Tor for a principal non-zero-divisor quotient>:
$$
\boxed{L_0F(M)=M/aM,\qquad L_1F(M)=\{m\in M:am=0\},\qquad L_iF(M)=0\quad(i\ge2).}
$$
These are naturally $B$-modules. For $M=A$, the degree-one group is zero; for $M=B$, multiplication by $a$ is zero and both degrees zero and one are $B$. Thus, when $B\ne0$, degree one explicitly detects the failure of this base-change <functor> to be exact.
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