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To expand \((1 - x + 2x^2)^6\) as far as the term in \(x^3\), we'll use the binomial expansion. The expansion of \((a + b + c)^n\) can be given by:

\[
(a + b + c)^n = \sum_{i+j+k=n} \frac{n!}{i!j!k!} a^i b^j c^k
\]

Here, \(a = 1\), \(b = -x\), \(c = 2x^2\), and \(n = 6\). We need to find the coefficients of the terms up to \(x^3\).

### Steps:
1. **Term 1 (constant term)**: When \(i = 6\), \(j = 0\), \(k = 0\):
   \[
   \text{Term} = \frac{6!}{6!0!0!} \cdot 1^6 \cdot (-x)^0 \cdot (2x^2)^0 = 1
   \]

2. **Term 2 (coefficient of \(x\))**: When \(i = 5\), \(j = 1\), \(k = 0\):
   \[
   \text{Term} = \frac{6!}{5!1!0!} \cdot 1^5 \cdot (-x)^1 \cdot (2x^2)^0 = 6(-x) = -6x
   \]

3. **Term 3 (coefficient of \(x^2\))**: There are two possibilities here:
   - \(i = 4\), \(j = 2\), \(k = 0\):
     \[
     \text{Term} = \frac{6!}{4!2!0!} \cdot 1^4 \cdot (-x)^2 \cdot (2x^2)^0 = \frac{6 \times 5}{2} x^2 = 15x^2
     \]
   - \(i = 4\), \(j = 0\), \(k = 1\):
     \[
     \text{Term} = \frac{6!}{4!0!1!} \cdot 1^4 \cdot (-x)^0 \cdot (2x^2) = \frac{6 \times 5}{1} \cdot 2x^2 = 30x^2
     \]

   Summing both contributions:
   \[
   15x^2 + 30x^2 = 45x^2
   \]

4. **Term 4 (coefficient of \(x^3\))**: There are two possibilities here:
   - \(i = 3\), \(j = 3\), \(k = 0\):
     \[
     \text{Term} = \frac{6!}{3!3!0!} \cdot 1^3 \cdot (-x)^3 \cdot (2x^2)^0 = \frac{20}{6}(-x^3) = -20x^3
     \]
   - \(i = 3\), \(j = 1\), \(k = 1\):
     \[
     \text{Term} = \frac{6!}{3!1!1!} \cdot 1^3 \cdot (-x)^1 \cdot (2x^2) = \frac{20}{1} \cdot (-2x^3) = -40x^3
     \]

   Summing both contributions:
   \[
   -20x^3 - 40x^3 = -60x^3
   \]

### Final Expression:
\[
(1 - x + 2x^2)^6 = 1 - 6x + 45x^2 - 60x^3 + \dots
\]

So, the blanks in the given expression should be filled as:

1. \(1\)
2. \(6\)
3. \(45\)
4. \(60\)

---

Would you like more details or have any questions? Here are some related questions you could explore:

1. How would the expansion change if the power were 5 instead of 6?
2. What is the general formula for expanding \((a + b + c)^n\)?
3. Can you expand \((1 - x + 2x^2)^6\) up to the term in \(x^4\)?
4. What is the binomial theorem, and how does it apply to this expansion?
5. How does the sign of the coefficient of \(x^3\) change if we switch the signs in the original expression?

**Tip**: The binomial expansion is a powerful tool for approximating polynomial expressions and plays a crucial role in many areas of mathematics, including calculus and probability theory.

Expand (1 - x + 2x^2)^6 as far as the term in x^3. Fill in the blanks.

Expand the polynomial expression (1 - x + 2x^2)^6 up to the term in x^3. This advanced algebra problem covers key concepts in binomial expansion and the binomial theorem, providing a step-by-step solution. Suitable for students in Grades 11-12.

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