I am following the Bayesian Methods for Machine Learning course on Coursera. Unfortunately, it glosses over many details, and I am struggling to understand how to check if a distribution is a conjugate of another.

For example, I would like to check if $ N(\mu_1 |m, s^2)$ prior is conjugate over parameter $ \mu_1$ for the likelihood $ N(X| \mu_1, \sigma_1^2)$.

As far as I understand, we have to multiply the two normals and check if the result is in the same family as the prior. Since we only want to check how the relevant variables behave, we can use a simplified form:

$ N(X| \mu_1, \sigma_1^2)$$ N(\mu_1 |m, s^2) \propto exp(-\frac{(x-\mu)^2}{2\sigma^2}) exp(-\frac{(\mu -m)^2}{2s^2})$

First of all, can I simplify and move the denominator in the $exp$ outside in this way?

$ exp(-b(x-\mu)^2) exp(-c(\mu -m)^2)$

where b and c are just two constant? Then, how would I proceed? Many thanks to whoever could clarify this.

$ $