To factor the quadratic expression \( 49p^2 - 7p - 30 \), we'll follow these steps:

### Step 1: Multiply the coefficient of \( p^2 \) by the constant
We need to multiply the first term (coefficient of \( p^2 \)) and the constant term:

\[
49 \times (-30) = -1470
\]

### Step 2: Find two numbers that multiply to -1470 and add to -7
We need to find two numbers that multiply to -1470 and add to -7. After checking possible factors, we find:

\[
35 \times -42 = -1470 \quad \text{and} \quad 35 + (-42) = -7
\]

### Step 3: Rewrite the middle term using these two numbers
We can now rewrite the middle term of the quadratic expression using the two numbers we found:

\[
49p^2 - 42p + 35p - 30
\]

### Step 4: Factor by grouping
Now, we'll group the terms and factor each group:

\[
(49p^2 - 42p) + (35p - 30)
\]

Factor out the greatest common factor from each group:

\[
7p(7p - 6) + 5(7p - 6)
\]

### Step 5: Factor the common binomial
We see that \( 7p - 6 \) is a common factor:

\[
(7p - 6)(7p + 5)
\]

### Final Answer:
The factored form of \( 49p^2 - 7p - 30 \) is:

\[
(7p - 6)(7p + 5)
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you factor more complex quadratic expressions?
2. What is the quadratic formula and how can it be used to solve this problem?
3. Can we apply factoring to higher-degree polynomials?
4. What is the difference between factoring by grouping and completing the square?
5. How does factoring relate to solving quadratic equations?

### Tip:
Always check your factorization by expanding the factors to ensure they give back the original expression!

Learn how to factor the quadratic expression 49p²-7p-30 with a detailed step-by-step solution. This guide covers key algebraic concepts such as factoring by grouping, the product-sum method, and related theorems. Suitable for students in Grades 9-10.

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