1. **State the problem:** Simplify the expression $$(-9x^2 + 2) \cdot 4x^2 + 2x^2 (5x^2 - 8) - (-7x^4 + 5x^2) \cdot 4$$. 2. **Apply the distributive property:** Multiply each term inside the parentheses by the factors outside. $$(-9x^2 + 2) \cdot 4x^2 = -9x^2 \cdot 4x^2 + 2 \cdot 4x^2 = -36x^{4} + 8x^{2}$$ $$2x^2 (5x^2 - 8) = 2x^2 \cdot 5x^2 - 2x^2 \cdot 8 = 10x^{4} - 16x^{2}$$ $$(-7x^4 + 5x^2) \cdot 4 = -7x^4 \cdot 4 + 5x^2 \cdot 4 = -28x^{4} + 20x^{2}$$ 3. **Rewrite the expression with these results:** $$-36x^{4} + 8x^{2} + 10x^{4} - 16x^{2} - (-28x^{4} + 20x^{2})$$ 4. **Simplify the subtraction of the last term:** $$-36x^{4} + 8x^{2} + 10x^{4} - 16x^{2} + 28x^{4} - 20x^{2}$$ 5. **Combine like terms:** For $x^{4}$ terms: $$-36x^{4} + 10x^{4} + 28x^{4} = (-36 + 10 + 28)x^{4} = 2x^{4}$$ For $x^{2}$ terms: $$8x^{2} - 16x^{2} - 20x^{2} = (8 - 16 - 20)x^{2} = -28x^{2}$$ 6. **Final simplified expression:** $$2x^{4} - 28x^{2}$$ 7. **Factor out common terms if desired:** $$2x^{2}(x^{2} - 14)$$