1. State the problem: Simplify the expression $4x^2 - 7x^2 + 3x - 5x^3$. 2. Formula and rules: To combine like terms use $ax^n + bx^n = (a+b)x^n$. 3. Identify like terms: The like terms are $4x^2$ and $-7x^2$, while $-5x^3$ and $3x$ are unlike and remain separate. 4. Combine the $x^2$ terms: $$4x^2 - 7x^2 = (4-7)x^2 = -3x^2$$ 5. Rewrite the expression grouping results: $$4x^2 - 7x^2 + 3x - 5x^3 = -3x^2 + 3x - 5x^3$$ 6. Rearrange in descending powers and present the final simplified form: $$-5x^3 - 3x^2 + 3x$$ Final answer: $-5x^3 - 3x^2 + 3x$.