Calculate expectation of stochastic integrals

Question

I am trying to calculate $$\mathbb{E}\left[\int^t_0 e^{\lambda s}dB_s \int^{t+h}_0 e^{\lambda s}dB_s \right], $$ where $(B_t)_{t\geq 0}$ is a brownian motion, $h>0$ and $\lambda > 0$ is some constant. My attempt was to write $$\int^t_0 e^{\lambda s}dB_s = \int^{t+h}_0 \mathbb{1}_{0\leq s\leq t}e^{\lambda s}dB_s$$ and use Ito's isometry: $$\mathbb{E}\left[\int^{t+h}_0 \mathbb{1}_{0\leq s\leq t}e^{\lambda s}dB_s \int^{t+h}_0 e^{\lambda s}dB_s\right]$$ $$=\mathbb{E}\left[\int^{t+h}_0 \mathbb{1}_{0\leq s \leq t}e^{2\lambda s}ds \right],$$ but i am not sure if this is correct.