1. **State the problem:** Simplify the expression $$(-2 x^2) (x^2)^3 + (2 x^2)^4 - (-2 x)^3 (+ 3 x^2) + (5 x)^2 (-x)^3$$. 2. **Recall exponent rules:** - $(a^m)^n = a^{m \cdot n}$ - $a^m \cdot a^n = a^{m+n}$ - $(-a)^n = (-1)^n a^n$ 3. **Simplify each term:** - First term: $$(-2 x^2)(x^2)^3 = (-2 x^2)(x^{2 \cdot 3}) = (-2 x^2)(x^6) = -2 x^{2+6} = -2 x^8$$ - Second term: $$(2 x^2)^4 = 2^4 (x^2)^4 = 16 x^{8}$$ - Third term: $$- (-2 x)^3 (3 x^2) = - \left((-1)^3 2^3 x^3\right)(3 x^2) = - (-1)(8 x^3)(3 x^2) = - (-24 x^{3+2}) = - (-24 x^5) = 24 x^5$$ - Fourth term: $$(5 x)^2 (-x)^3 = 5^2 x^2 \cdot (-1)^3 x^3 = 25 x^2 (-1) x^3 = -25 x^{2+3} = -25 x^5$$ 4. **Combine all terms:** $$-2 x^8 + 16 x^8 + 24 x^5 - 25 x^5 = ( -2 + 16 ) x^8 + (24 - 25) x^5 = 14 x^8 - x^5$$ 5. **Final answer:** $$\boxed{14 x^8 - x^5}$$