$$\begin{array}{ll|l}
A &= (a+b)^*\;ab\;(a+b)^* & \text{Strings containing substring } ``ab"\\[1em]
A^R &= (a+b)^*\;ba\;(a+b)^* & \text{Strings containing substring } ``ba"\\[2em]
B &= a^*b^* & \text{All $a$'s should come before any $b$'s}\\[1em]
B^R &= b^*a^* & \text{All $b$'s should come before any $a$'s}\\[2em]
C &= (a+b)^* & \text{All strings}
\end{array}$$
Now,
$A+B$ doesn't give us all the strings. For example, $A+B$ doesn't contain the string $``ba"$.
Thus, $A+B \neq C$
$A^R+B^R$ doesn't give us all the strings. For example, $A^R+B^R$ doesn't contain the string $``ab"$.
Thus, $A^R+B^R \neq C$
$A^R+B$ does give us all the strings!
First, lets look at which strings don't come in $B$. They are the ones that contain atleast one such $``a"$ that occurs after some $``b"$. For example, $B$ doesn't contain the string $``aba"$
Now, we observe that all the strings that do not come in $B$, satisfy the property for $A^R$, and hence come in $A^R$.
Thus, if we union $A^R$ and $B$, we will get all strings.
Hence, $A^R + B = C$
Another way to verify it is by constructing an NFA for $A^R + B$, converting it to DFA and minimizing it.