To factor the quadratic expression \(7c^2 - 50c + 7\), we can apply the same factoring by grouping technique.

### Step 1: Multiply the coefficient of \(c^2\) (which is 7) by the constant term (which is also 7):
\[
7 \times 7 = 49
\]

### Step 2: Find two numbers that multiply to 49 and add to \(-50\):
The numbers are \(-49\) and \(-1\), since:
\[
-49 \times -1 = 49 \quad \text{and} \quad -49 + (-1) = -50
\]

### Step 3: Rewrite the middle term \(-50c\) using \(-49c\) and \(-1c\):
\[
7c^2 - 49c - 1c + 7
\]

### Step 4: Factor by grouping:
\[
7c(c - 7) - 1(c - 7)
\]

### Step 5: Factor out the common binomial \((c - 7)\):
\[
(c - 7)(7c - 1)
\]

So, the factored form of \(7c^2 - 50c + 7\) is:
\[
\boxed{(c - 7)(7c - 1)}
\]

Would you like further details or have any questions?

Here are some related questions you might find helpful:

1. How do you determine the factors of a quadratic expression quickly?
2. What is the difference between factoring and expanding expressions?
3. How can you verify the factors of a quadratic equation?
4. Can you factor quadratics where the coefficients are not integers?
5. How does the discriminant of a quadratic equation relate to its factors?

**Tip:** Always check your factored form by expanding it back out to ensure it matches the original quadratic expression.

Learn how to factor the quadratic expression 7c^2 - 50c + 7 with a detailed step-by-step solution. This problem involves factoring by grouping and is suitable for advanced algebra students in Grades 9-12.

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