Let $w(x_1, . . . , x_n)$ be a loop term, and let $A$ be a normal subloop of $Q$. Prove that $w(a_1A, · · · , a_nA) = w(a_1, . . . , a_n)A$ for every $a_1, . . . , a_n \in A$.
A subloop $A$ of $Q$ is normal if for all $x,y \in Q$ $xA=Ax$, $x(yA) = (xy)A$, and $(Ax)y=A(xy)$. So I'm not sure what exactly needs to be done here. It seems immediate if we can always reassociate the $a_i$ and then push them all to the other side of $A$ to get the resulting term.
Recall that the definition of term is recursive:
A variable symbol $x$ is a term, and
if $f$ and $g$ are terms, then $f\cdot g$ is a term.
Whenever a collection is defined recursively, you need to use induction to prove statements about that collection. So prove your statement for $w(x_1,\dots,x_n)=x_i$ and then, assuming your statement is true for $w_1$ and $w_2$, prove it is true for $w_1\cdot w_2$