To factorize the expression $3\sin^2\theta - 12\sin\theta + 9$, we can follow these steps:

1. Identify the expression:

   The given expression is a quadratic expression in terms of $\sin\theta$.

2. Factor out the greatest common factor (GCF):

   We see that the GCF of all the terms is $3$, so we factor that out: $3(\sin^2\theta - 4\sin\theta + 3)$

3. Factorize the quadratic expression:

   Now, we need to factor the quadratic expression inside the parentheses: $\sin^2\theta - 4\sin\theta + 3$.

   We look for two numbers that multiply to $3$ (the constant term) and add to $-4$ (the coefficient of $\sin\theta$).

   These numbers are $-3$ and $-1$, so we can factorize the quadratic as: $(\sin\theta - 3)(\sin\theta - 1)$

4. Combine the factorizations:

   Putting it all together, we have: $3(\sin\theta - 3)(\sin\theta - 1)$

Thus, the factorized form of $3\sin^2\theta - 12\sin\theta + 9$ is:

$3(\sin\theta - 3)(\sin\theta - 1)$

Would you like more details on any of these steps or have any questions?

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Here are 8 related questions you might find interesting:

1. How do you solve quadratic equations by factoring?
2. What are the different methods to solve trigonometric equations?
3. How do you factor a quadratic expression when the leading coefficient is not 1?
4. What is the significance of the discriminant in quadratic equations?
5. How do you apply the quadratic formula in trigonometric equations?
6. What is the connection between factoring and solving quadratic equations?
7. How do you simplify trigonometric expressions?
8. What is the role of common factors in simplifying expressions?

Tip: Always check your factorization by expanding the factors back out to ensure they match the original expression.