= Solution
For a homogeneous canonical scalar, $\rho_\phi=\dot\phi^2/2+V$ and $P_\phi=\dot\phi^2/2-V$. The acceleration equation requires $V>\dot\phi^2$ for <cosmic inflation>, equivalently $\epsilon_H=-\dot H/H^2<1$ in the flat scalar-dominated limit. Inflation need not be exactly de Sitter.
A standard sufficient regime for a long period is positive <potential energy> dominating the <kinetic energy>, physical <gradient> <energy> and <curvature>. Homogeneity over an expanding approximately Hubble-sized patch allows the <gradient> term to <redshift> away. In comoving coordinates the <gradient> <energy> is $|\nabla_{\rm com}\phi|^2/(2a^2)$; the <gradient> in the displayed <energy> formula must be interpreted physically if no $a^{-2}$ is written there. Require
$$
\frac{\dot\phi^2}{2}\ll V,\quad\frac{|\nabla_{\rm com}\phi|^2}{2a^2}\ll V,\quad\frac{|k|}{a^2}\ll\frac{8\pi V}{3m_{\rm pl}^2},\quad|\ddot\phi|\ll3H|\dot\phi|.
$$
The slow-roll equations are then $3H\dot\phi\simeq-V'$ and $H^2\simeq8\pi V/(3m_{\rm pl}^2)$. Sufficient potential conditions are
$$
\epsilon_V=\frac{m_{\rm pl}^2}{16\pi}\left(\frac{V'}V\right)^2\ll1,\qquad |\eta_V|=\left|\frac{m_{\rm pl}^2}{8\pi}\frac{V''}V\right|\ll1.
$$
The potential must remain sufficiently flat over a large enough field interval to produce the required expansion. These slow-roll conditions are stronger than the bare necessary condition for acceleration.
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