1. **State the problem:** Add the polynomials $$-8x^3y^2 + 6x^2y - 6x + 9$$ and $$7x^3y^2 + 9x - 4$$ using vertical form and simplify completely. 2. **Write the polynomials aligned by like terms:** $$\begin{array}{r} -8x^3y^2 + 6x^2y - 6x + 9 \\ +7x^3y^2 \quad\quad + 9x - 4 \end{array}$$ 3. **Add like terms:** - For $$x^3y^2$$ terms: $$-8x^3y^2 + 7x^3y^2 = (-8 + 7)x^3y^2 = -1x^3y^2 = -x^3y^2$$ - For $$x^2y$$ term: only $$6x^2y$$ (no like term in second polynomial) - For $$x$$ terms: $$-6x + 9x = (-6 + 9)x = 3x$$ - For constants: $$9 - 4 = 5$$ 4. **Write the simplified sum:** $$-x^3y^2 + 6x^2y + 3x + 5$$ **Final answer:** $$-x^3y^2 + 6x^2y + 3x + 5$$