= Solution
Disconnect the <amide> into \b[2-propylhexanoic acid and N-methyl-2-(4-hexylphenyl)ethylamine]. This keeps the <nitrogen> substituents distinct: one is Me and the other is the two-carbon aryl side chain. The acid's alpha <carbon> must bear both n-butyl and n-propyl groups.
For the acid fragment, esterify the supplied <hexanoic acid>, form its <ester enolate> with LDA, and alkylate with 1-iodopropane:
$$
\mathrm{EtO_2CCH_2(nBu)}
\longrightarrow\mathrm{EtO_2CCH(nPr)(nBu)}.
$$
<hydrolysis> gives 2-propylhexanoic acid; conversion with $\mathrm{SOCl_2}$ gives the <acyl chloride> for the final coupling. Limit the alkylating reagent and isolate the monoalkylated <ester> to avoid overalkylation.
For the aryl fragment, use another portion of <hexanoic acid> to make hexanoyl chloride. A <Friedel-Crafts acylation> of <bromobenzene> with this reagent and $\mathrm{AlCl_3}$ gives ortho and para products; isolate the para bromoaryl <ketone>. A <Wolff-Kishner reduction> converts $\mathrm{ArCO(CH_2)_4Me}$ into $\mathrm{ArCH_2(CH_2)_4Me}$ without deliberately hydrogenolyzing the aryl bromide. Acylation followed by <carbonyl reduction> installs an unrearranged n-hexyl group; direct Friedel–Crafts alkylation is a less reliable way to do that.
Convert the retained aryl bromide into its <grignard reagent>, open <ethylene oxide>, and work up to obtain $\mathrm{p\!-\!HexC_6H_4CH_2CH_2OH}$. Oxidize the primary <alcohol> to $\mathrm{p\!-\!HexC_6H_4CH_2CHO}$ under conditions that stop at the <aldehyde>, then perform <reductive amination> with <methylamine> and <sodium triacetoxyborohydride>. Excess <methylamine> helps favour the desired secondary amine over further alkylation. Finally acylate that secondary amine with the isolated <acyl chloride>, using a separate base to capture HCl.
\b[The final <amide> has an N-methyl group, the para n-hexyl phenethyl group, and the required n-butyl/n-propyl acyl branching.] The <carbon> check is $9$ in the acyl fragment plus $1+2+6+6=15$ in the amine fragment, giving $24$ overall. No absolute <stereochemistry> is specified at the acid alpha <carbon>.
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