1. **State the problem:** Solve and verify the equality $$(x\sqrt{3})^2 = (-\sqrt{1.5} x)^2$$ and simplify both sides. 2. **Recall the formula:** The square of a product is the product of the squares: $$(ab)^2 = a^2 b^2$$. 3. **Simplify the left side:** $$ (x\sqrt{3})^2 = x^2 \times (\sqrt{3})^2 = x^2 \times 3 = 3x^2 $$ 4. **Simplify the right side:** $$ (-\sqrt{1.5} x)^2 = (-1)^2 \times (\sqrt{1.5})^2 \times x^2 = 1 \times 1.5 \times x^2 = 1.5 x^2 $$ 5. **Check the equality:** The left side is $3x^2$ and the right side is $1.5 x^2$, so they are not equal unless $3x^2 = 1.5 x^2$. 6. **Solve for $x$ if equality holds:** $$ 3x^2 = 1.5 x^2 $$ Divide both sides by $x^2$ (assuming $x \neq 0$): $$ \cancel{3} \cancel{x^2} = \cancel{1.5} \cancel{x^2} $$ which simplifies to $$ 3 = 1.5 $$ which is false. 7. **Conclusion:** The original equality holds only if $x=0$ because then both sides equal zero. For any other $x$, the equality does not hold. **Final answer:** The equality $$(x\sqrt{3})^2 = (-\sqrt{1.5} x)^2$$ simplifies to $$3x^2 = 1.5 x^2$$ which is true only if $x=0$.