how do i synthesize (E) 2-methyl-1-phenylpent-1-en-3-one?
Looks like a crossed aldol condensation between benzaldehyde and pentan-3-one.

In a classical approach, you would (under stirring)


Add your base to a dry solvent of your choice
Cool down (ice bath or dry ice/isopropanol)
Add pentan-3-one dropwise to make sure that it's all immediately converted to the corresponding enolate. You do not want self-condensation of the ketone!
Let it come to room temperature (and stir for a while)
Cool down again
Slowly add benzaldehyde. Freshly distilled benzaldehyde is recommended.
Let it come to room temperature again (and stir for a another while)


Check the progress of the reaction by TLC. You should have at least the aldol addition product by now. In order to facilitate elimination of water and thus complete the aldol condensation, it might be necessary to acidify the mixture and heat it.

There are other more modern (e.g. Mukayama aldol reaction, using the silyl enol ether of the ketone) and/or "greener" methods, but they are mostly a variation of the basic approach described above.