How do you factor the expression x^3 + 7x^2 + 5x + 35 by grouping, and what is the resulting expression?

To factor the expression x³ + 7x² + 5x + 35 using the grouping method, we can follow these steps:

1. First, we should group the terms in pairs:

   (x³ + 7x²) + (5x + 35)

2. Next, we factor out the greatest common factor (GCF) from each group:

   From the first group (x³ + 7x²), the GCF is x², so:

   x²(x + 7)

3. From the second group (5x + 35), the GCF is 5, so:

   5(x + 7)

4. Now, you can rewrite the expression using these factored forms:

   So far, we have:

   x²(x + 7) + 5(x + 7)

5. Now notice that both terms have a common factor of (x + 7):

   Factoring out (x + 7) gives us:

   (x + 7)(x² + 5)

Therefore, the expression x³ + 7x² + 5x + 35 factors to:

(x + 7)(x² + 5)

This is the resulting expression after using the grouping method to factor.