The first search for nonresonant production of Higgs boson pairs (HH) with one H decaying into four leptons and the other into a pair of b quarks is presented, using proton-proton collisions recorded at a center-of-mass energy of $\sqrt{s} =$ 13 TeV by the CMS experiment. The analyzed data correspond to an integrated luminosity of 138 fb$^{-1}$. A 95% confidence level upper limit of 32.4 is set on the signal strength modifier $\mu$, defined as the ratio of the observed HH production rate in the ${\mathrm{H}\mathrm{H}} \to\mathrm{Z}\mathrm{Z}^{*}\mathrm{b}\mathrm{\bar{b}}\to 4\ell\mathrm{b}\mathrm{\bar{b}}$ decay channel to the standard model expectation. Possible modifications of the H trilinear coupling $\lambda_\text{HHH}$ with respect to the standard model (SM) value are investigated. The coupling modifier $\kappa_{\lambda}$, defined as $\lambda_\text{HHH}$ divided by its SM prediction, is constrained to be within the observed (expected) range $-$8.8 ($-$9.8) $< \kappa_{\lambda} <$ 13.4 (15.0) at 95% confidence level.

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