If there was an inner product between two vectors, say:
$\langle x , a \rangle$
with $x$ being a variable and $a$ a constant. Would the following equality be valid?
$\langle x , a \rangle b = \langle xb,a\rangle = x\langle b,a \rangle$
Assuming linear properties, I would think this is correct, but I'm not sure with $x$ being variable.
Your inner product is defined like this $$ \langle ., . \rangle : V \times V \to F $$ for some vector space $V$ over some field $F$, two vectors from $V$ get a scalar from $F$ assigned.
Regarding your first equation: $$ \langle x , a \rangle b = \langle xb,a\rangle $$
A general inner product over complex numbers is $$ \langle x, y \rangle = y^+ M x $$ where $+$ is adjugation (not sure if this a correct English term), thus transposition and complex conjugation, and $M$ is some Hermitian positive-definite matrix, typically the identity matrix.
So $$ \langle x , a \rangle b = (a^+ M x) b = a^+ M (xb) = \langle xb,a\rangle $$ So your equality is true, where $x, a$ should be vectors from $V$, $b$ a scalar from $F$.
Regarding your second equation:
The second equation $$ \langle xb,a\rangle = x\langle b,a \rangle $$ is only true if $V = F$, otherwise $\langle b,a \rangle$ makes no sense.