1. State the problem. Simplify $\left( 6x^2 y^3 + x^3 - 5x^3 y^2 \right) - \left( 2x^3 y^2 - 7y^2 + 4x^3 \right)$. 2. Use the subtraction rule. Subtracting a parentheses means distribute the minus sign to every term inside the second parentheses: $$A-B=A+(-B).$$ 3. Distribute the minus sign. $$\left(6x^2y^3+x^3-5x^3y^2\right)-\left(2x^3y^2-7y^2+4x^3\right)$$ $$=6x^2y^3+x^3-5x^3y^2-2x^3y^2+7y^2-4x^3.$$ 4. Combine like terms. - Terms with $x^3y^2$: $-5x^3y^2-2x^3y^2=-7x^3y^2$ - Terms with $y^2$: $7y^2$ - Terms with $x^3$: $x^3-4x^3=-3x^3$ So the expression becomes: $$6x^2y^3-7x^3y^2+7y^2-3x^3.$$ 5. Final simplified answer. $$\boxed{6x^2y^3-7x^3y^2+7y^2-3x^3}$$