The expression $(x - 3)^2 \cdot (x + 1)^3$ is already in factored form. If you want to expand it, you can use the binomial theorem for each part.

Let’s expand this step-by-step:

Step 1: Expand $(x - 3)^2$

$(x - 3)^2 = x^2 - 6x + 9$

Step 2: Expand $(x + 1)^3$

$(x + 1)^3 = x^3 + 3x^2 + 3x + 1$

Step 3: Multiply the two expanded forms

Now we have: $(x - 3)^2 \cdot (x + 1)^3 = (x^2 - 6x + 9)(x^3 + 3x^2 + 3x + 1)$

Expanding this product term-by-term:

1. $x^2 \cdot x^3 = x^5$

2. $x^2 \cdot 3x^2 = 3x^4$

3. $x^2 \cdot 3x = 3x^3$

4. $x^2 \cdot 1 = x^2$

5. $-6x \cdot x^3 = -6x^4$

6. $-6x \cdot 3x^2 = -18x^3$

7. $-6x \cdot 3x = -18x^2$

8. $-6x \cdot 1 = -6x$

9. $9 \cdot x^3 = 9x^3$

10. $9 \cdot 3x^2 = 27x^2$

11. $9 \cdot 3x = 27x$

12. $9 \cdot 1 = 9$

Step 4: Combine all terms

Now add together all terms by degree: $x^5 + (3x^4 - 6x^4) + (3x^3 - 18x^3 + 9x^3) + (x^2 - 18x^2 + 27x^2) + (-6x + 27x) + 9$

Simplifying: $= x^5 - 3x^4 - 6x^3 + 10x^2 + 21x + 9$

So, the expanded form is: $(x - 3)^2 \cdot (x + 1)^3 = x^5 - 3x^4 - 6x^3 + 10x^2 + 21x + 9$

Would you like more details on any step?

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Related Questions:

1. How do we expand binomials raised to a power using the binomial theorem?
2. How do we multiply polynomials systematically to avoid missing terms?
3. What are common mistakes to avoid when expanding products of polynomials?
4. How can we verify the expansion result by factoring it back?
5. How does expanding polynomials help in calculus, especially for differentiation?

Tip:

When expanding products of polynomials, it's helpful to align terms by their degree to avoid errors and simplify addition afterward.