Okay, let's see. I need to expand and simplify this expression: x times (x times (1 plus x) plus 2x) minus 3 times (x squared minus x plus 2). Hmm. Alright, let me start by breaking it down step by step. First, maybe I should handle the innermost parentheses. The expression inside the first big parenthesis is x(1 + x) + 2x. Let me compute that part first.

So, x(1 + x) would be x*1 + x*x, which is x + x². Then adding 2x to that. So x + x² + 2x. Combining like terms, x + 2x is 3x, so that becomes x² + 3x. Okay, so the first part simplifies to x² + 3x. Then, the entire expression is x multiplied by that result. So x*(x² + 3x). Let me compute that. Multiplying x by each term inside the parentheses. x*x² is x³, and x*3x is 3x². So that gives x³ + 3x². Alright, so the first part of the original expression simplifies to x³ + 3x².

Now, moving on to the second part of the original expression: minus 3 times (x² - x + 2). Let's distribute the -3 across each term inside the parentheses. So, -3*x² is -3x², -3*(-x) is +3x, and -3*2 is -6. So that part becomes -3x² + 3x - 6.

Now, combining both parts: the first part is x³ + 3x², and the second part is -3x² + 3x - 6. Let's write that out: x³ + 3x² - 3x² + 3x - 6. Now, let's combine like terms. The 3x² and -3x² cancel each other out, leaving x³ + 3x - 6.

Wait, let me check that again. The first term is x³, then 3x² minus 3x² is zero, so those cancel. Then we have +3x and -6. So yes, combining those gives x³ + 3x - 6. Hmm. Let me verify each step again to make sure I didn't make a mistake.

Starting with the innermost parentheses: x(1 + x) is x + x². Then adding 2x gives x + x² + 2x = x² + 3x. Multiplying by x gives x³ + 3x². Then the second part: -3 times (x² - x + 2) is -3x² + 3x -6. Adding those together: x³ + 3x² -3x² +3x -6. The 3x² and -3x² indeed cancel, leaving x³ +3x -6. That seems right.

Wait, but let me check the signs again. In the second part, it's minus 3 times the entire parenthesis. So the signs inside the parenthesis should be distributed correctly. Original expression is x*(...) -3*(...), so the second part is -3*(x² -x +2). So that's -3x² +3x -6. Yes, correct. So when we combine with the first part, which is x³ +3x², then subtract 3x², etc. So yes, the final simplified expression is x³ +3x -6.

I think that's it. Let me just go through each step once more to ensure there's no arithmetic error.

First part: x*(x*(1+x) +2x). Let's compute the innermost bracket again: x*(1+x) is x +x². Adding 2x gives x +x² +2x =x² +3x. Multiply by x: x³ +3x². Correct.

Second part: -3*(x² -x +2) = -3x² +3x -6. Correct.

Combine the two parts: x³ +3x² -3x² +3x -6. The 3x² and -3x² cancel, so x³ +3x -6. Yep, that's right. So the simplified form is x³ +3x -6. I don't think there are any like terms left to combine. The exponents are all different: x³, x, and constant. So that's the final answer.
Therefore, the expanded and simplified form of the given expression is $\boxed{x^3 + 3x - 6}$.
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