A few disconnection strategies for the retrosynthesis of this conjugated ketone. This video shows how you can split up a 1,4-diX functional group relationship in different ways that use umpolung chemistry.
There is an enone (alpha,beta-unsaturated ketone) in the molecule and so that is a great place for a disconnection to start a retrosynthetic analysis. This type of C=C bond can be constructed typically by an aldol condensation or Wittig reaction - here the intramolecular aldol condensation will be the simplest and form a 5-membered ring in preference to a 3-membered ring. This takes you back to a 1,4-dicarbonyl species, which is a flag for using reagents associated with umpolung chemistry (reversed polarity reagents).
There is a branch point in the molecule so disconnecting around that will simplify the reactants for the next step of the retrosynthesis.
One option is to use a nitroalkane as a d1 synthon, as a sodt nucleophile at carbon once deprotonated. The anion of the nitroalkane can do Michael addition with another enone. This is a strategy that can work because you can use a Nef reaction to convert the nitro group into a carbonyl group later on. The Nef reaction uses titanium trichloride (TiCl3) and water.
An alternative strategy involves using an enamine, which has the advantage of disconnecting back to a symmetrical starting material (pentan-3-one). An enamine is a soft nucleophile that reacts cleanly with good soft electrophiles, such as in an SN2 reaction that has a stabilised transition state (here an adjacent carbonyl group).
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