Hint: I won't give a full answer since this is a very specific question, but I can give you a long outline here. A Lagrangian is set-up with an objective function (in this case utility) plus or minus (depending on Kuhn Tucker constraints, which we won't worry about) a constraint function (in this case, income) which is set to zero and then multiplied by the Lagrange multiplier, which we will denote $\lambda$.
So we set up the Lagrangian function. (Plug in the utility function yourself.)
$$\mathcal{L} = u(x_1, x_2, x_3) - \lambda (p_1x_1 + p_2x_2 + p_3x_3 - m)$$
Solve for the partial derivatives $\frac{\partial \mathcal{L}}{\partial x_1}$, $\frac{\partial \mathcal{L}}{\partial x_2}$, and $\frac{\partial \mathcal{L}}{\partial x_3}$ and set them equal to zero (You can also solve for the partial of $\lambda$, but it just gives you back the budget constraint, which I skip to). For example, you get for your first derivative:
$$\frac{\partial \mathcal{L}}{\partial x_1} = \alpha (x_1 + z_1)^{\alpha - 1} - \lambda p_1 = 0$$
Once you solve the three partials, the easiest thing for you to do is solve for $\lambda$ with respect to the other variables for all three partials, and then figure out how to substitute them into the budget constraint. You should solve for each $x$ in terms of the prices, maybe some $z$'s, the greek letters $(\alpha, \beta, \gamma)$ and income $m$. (I tried working some of it out; enjoy your algebra.)