Here's how to balance the chemical equation for the reaction of aqueous hydrochloric acid with solid manganese(IV) oxide:

Unbalanced Equation:

HCl(aq) + MnO₂(s) → MnCl₂(aq) + H₂O(l) + Cl₂(g)

Balancing the Equation:

1. Manganese (Mn): The Mn is balanced on both sides (1 on each).

2. Chlorine (Cl): There are 3 Cl atoms on the product side (2 in MnCl₂ and 1 in Cl₂) and only 1 on the reactant side. We need to add a coefficient of 4 in front of HCl.

3. Oxygen (O): There are 2 O atoms on the reactant side and 1 on the product side. We need to add a coefficient of 2 in front of H₂O.

4. Hydrogen (H): With the new coefficients, there are 8 H atoms on the reactant side and 4 on the product side. We'll adjust the coefficient in front of H₂O to 4.

Balanced Equation:

4HCl(aq) + MnO₂(s) → MnCl₂(aq) + 2H₂O(l) + Cl₂(g)