To factor the quadratic expression \(5x^2 + 32x + 35\), we need to find two numbers that multiply to \(5 \times 35 = 175\) and add up to \(32\). 

Let's follow the steps:

1. **Identify coefficients:**
   - \(a = 5\), \(b = 32\), \(c = 35\).

2. **Find two numbers that multiply to \(a \times c = 175\) and add up to \(b = 32\):**
   - The numbers are \(7\) and \(25\) because \(7 \times 25 = 175\) and \(7 + 25 = 32\).

3. **Rewrite the middle term using these two numbers:**
   \[
   5x^2 + 7x + 25x + 35
   \]

4. **Factor by grouping:**
   \[
   (5x^2 + 7x) + (25x + 35)
   \]
   \[
   x(5x + 7) + 5(5x + 7)
   \]

5. **Factor out the common factor \((5x + 7)\):**
   \[
   (x + 5)(5x + 7)
   \]

So, the factored form of \(5x^2 + 32x + 35\) is:

\[
(x + 5)(5x + 7)
\]

Would you like more details or have any questions? 

### Related Questions:
1. How do you factor a quadratic equation using the quadratic formula?
2. How can you determine if a quadratic expression is factorable by simple inspection?
3. What are the applications of factoring quadratic equations in real-world problems?
4. How do you factor expressions with coefficients that are prime numbers?
5. What are other methods for factoring polynomials besides grouping?

### Tip:
Always double-check your factorization by expanding the factored form to ensure it matches the original expression.

Learn how to factor the quadratic expression 5x^2 + 32x + 35 step by step. This tutorial covers important algebra concepts such as polynomial factorization and the factorization theorem, suitable for students in Grades 8-10.

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