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To factor the quadratic expression \(5x^2 - 12x + 7\) by grouping, we can follow these steps:

### Step 1: Multiply the leading coefficient by the constant term.
The leading coefficient is 5, and the constant term is 7:
\[
5 \times 7 = 35
\]

### Step 2: Find two numbers that multiply to 35 and add to -12.
We are looking for two numbers that multiply to 35 and add up to -12. These numbers are -5 and -7 because:
\[
-5 \times -7 = 35 \quad \text{and} \quad -5 + (-7) = -12
\]

### Step 3: Rewrite the middle term using the two numbers found.
We can rewrite the expression as:
\[
5x^2 - 5x - 7x + 7
\]

### Step 4: Group the terms in pairs.
Now group the terms:
\[
(5x^2 - 5x) - (7x - 7)
\]

### Step 5: Factor out the greatest common factor from each group.
Factor each group:
\[
5x(x - 1) - 7(x - 1)
\]

### Step 6: Factor out the common binomial factor.
The common factor is \((x - 1)\), so we factor it out:
\[
(5x - 7)(x - 1)
\]

### Final Factored Form:
Thus, the factored form of \(5x^2 - 12x + 7\) is:
\[
(5x - 7)(x - 1)
\]

Would you like further details or explanations on any part of this process?

Here are some related questions to consider:
1. How can you verify that the factored form is correct?
2. What is the role of the distributive property in factoring?
3. How would the process change if the middle term was positive?
4. Can all quadratics be factored by grouping? If not, when does it fail?
5. How does factoring relate to solving quadratic equations?

**Tip:** Always check your factorization by expanding the factors to ensure you return to the original polynomial.

Factor the polynomial by grouping: 5x^2 - 12x + 7

Learn how to factor the quadratic polynomial 5x^2 - 12x + 7 by grouping. This solution covers key algebra concepts, including finding pairs of numbers that sum to the middle term and factoring out the greatest common factor. Suitable for students in Grades 9-10.

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