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The problem sheet you've uploaded includes a set of factorizations involving quadratic expressions. The goal is to insert the correct numbers (from 0 to 9, without repetition) in the empty boxes to make each equation correct.

Let’s walk through the four problems one by one:

1. **Equation 1:**
   \[
   x^2 - x - \square = (x + 2)(x - \square)
   \]
   We need to factor the quadratic expression on the left-hand side and match the constants accordingly.

2. **Equation 2:**
   \[
   x^2 - 1 + x + 1 = (x - 2)(x - \square)
   \]
   Simplify the left-hand side and solve by factoring, making sure the factors match.

3. **Equation 3:**
   \[
   x^2 + \square x + 1 = (x + \square)(x + 2)
   \]
   Again, this involves factoring the quadratic on the left-hand side, with the correct constants.

4. **Equation 4:**
   \[
   x^2 - \square x - 24 = (x - 6)(x + \square)
   \]
   Factor the quadratic and find the missing values.

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Let me help you with these step-by-step. Would you like me to go through the solutions in detail for each equation?

Also, here are 5 related questions you might explore:

1. How do you factor general quadratic equations?
2. What conditions are required for a quadratic to be factorable?
3. Can all quadratics be solved by factoring, or are there cases where another method is necessary?
4. What role does the discriminant play in determining factorability?
5. How can you quickly check if your factors are correct?

**Tip:** Always verify the product of the factors to ensure they match the original quadratic.

Using each digit from 0-9 only once, complete the following factorizations involving quadratic expressions.

Learn how to solve quadratic equations by factoring and filling in the blanks with digits from 0-9. This problem involves key algebraic concepts, such as the quadratic equation formula and factorization, suitable for students in Grades 7-10.

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