= Solution
Write $\int_{\mathbf p}=\int d^3\mathbf p/(2\pi)^3$. The <canonical commutation relations> and the product rule for a <commutator> give
$$
[a_{\mathbf p},a_{\mathbf q}^\dagger a_{\mathbf q}]=(2\pi)^3\delta^3(\mathbf p-\mathbf q)a_{\mathbf q},\qquad
[b_{\mathbf p}^\dagger,b_{\mathbf q}^\dagger b_{\mathbf q}]=-(2\pi)^3\delta^3(\mathbf p-\mathbf q)b_{\mathbf q}^\dagger.
$$
The other-species terms vanish. Integrating the <Dirac delta distributions> in the <Hamiltonian> therefore gives
$$
[a_{\mathbf p},H]=E_p a_{\mathbf p},\qquad [b_{-\mathbf p}^\dagger,H]=-E_p b_{-\mathbf p}^\dagger,
$$
where $E_{-p}=E_p$. Consequently the <complex scalar field> satisfies
$$
[\phi(t,\mathbf x),H]=\int_{\mathbf p}\frac{E_p}{\sqrt{2E_p}}\left(-b_{-\mathbf p}^\dagger e^{iE_pt}+a_{\mathbf p}e^{-iE_pt}\right)e^{i\mathbf p\cdot\mathbf x}.
$$
Direct <differentiation> of the time factors gives the same expression, proving the <Heisenberg equation of motion>
$$
\boxed{i\partial_t\phi(t,\mathbf x)=[\phi(t,\mathbf x),H].}
$$
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