= Solution
Disconnect the tertiary <alcohol> at its bond to the alkenyl group. The required <carbonyl> partner is $\mathrm{MeCOCH(Me)Et}$; the other partner is an \b[E-configured $\mathrm{PhCH_2CH{=}CH}$ organometallic reagent]. This disconnection fixes the <alkene> geometry before the <alcohol> is made.
First form the ethyl-side <enolate> of <butan-2-one> under equilibrating conditions, then trap it with ethyl iodide and isolate the required monoalkylated <ketone>:
$$
\mathrm{MeCOCH_2Me}\longrightarrow\mathrm{MeCOCH(Me)Et}.
$$
The incoming group is Et: methylation would give a <carbon> skeleton one <carbon> too short. Because an unsymmetrical <ketone> can form two <enolates> and undergo repeated alkylation, regioisomer separation and controlled conversion are part of this step.
Independently perform a halo-<Wittig reaction> on phenylacetaldehyde using $\mathrm{Ph_3P{=}CHBr}$ and isolate \b[E-$\mathrm{PhCH_2CH{=}CHBr}$] from any geometrical mixture. <magnesium> insertion in dry ether gives the corresponding <vinylmagnesium halide>; add the isolated <ketone> at low temperature and protonate the resulting <alkoxide>. Addition to C=O preserves the pre-existing C=C geometry:
$$
(E)\text{-}\mathrm{PhCH_2CH{=}CHMgBr}
+\mathrm{MeCOCH(Me)Et}
\longrightarrow
(E)\text{-}\mathrm{PhCH_2CH{=}CHC(OH)(Me)CH(Me)Et}.
$$
\b[The E <alkene> and the complete fifteen-carbon skeleton are obtained.] No absolute configuration is specified at the newly formed <alcohol> centre, so an unselected stereoisomeric mixture there is acceptable. An alternative <carbon>–<carbon> disconnection is <acetylide> addition to the same <ketone> followed by a controlled trans-selective <alkyne> reduction; the vinyl-reagent route avoids relying on reduction of a tertiary propargylic <alcohol>.
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