The problem asks to expand and fully simplify the given expression:

$(x^2 + 7x + 6)(x + 4)$

Step-by-step Solution:

1. Distribute each term in $(x + 4)$ across $(x^2 + 7x + 6)$: $(x^2 + 7x + 6)(x + 4) = (x^2 + 7x + 6) \cdot x + (x^2 + 7x + 6) \cdot 4$

2. Expand each term:

   • Multiply $(x^2 + 7x + 6)$ by $x$: $(x^2 + 7x + 6) \cdot x = x^3 + 7x^2 + 6x$
   • Multiply $(x^2 + 7x + 6)$ by $4$: $(x^2 + 7x + 6) \cdot 4 = 4x^2 + 28x + 24$

3. Combine the results: $x^3 + 7x^2 + 6x + 4x^2 + 28x + 24$

4. Group like terms:

   • $x^3$ (only term with $x^3$)
   • Combine $7x^2 + 4x^2 = 11x^2$
   • Combine $6x + 28x = 34x$
   • Constant term: $24$

5. Write the final simplified expression: $x^3 + 11x^2 + 34x + 24$

Final Answer:

$x^3 + 11x^2 + 34x + 24$

Would you like further details or have questions about any step? Here are some related questions for practice:

1. How do you factorize the resulting cubic polynomial $x^3 + 11x^2 + 34x + 24$?
2. Can you verify the solution by substituting specific values of $x$?
3. What is the degree of the polynomial after expansion?
4. How do you handle similar problems with more terms, such as $(x^2 + 7x + 6)(x + 4 + y)$?
5. How can you apply these methods to real-world problems involving polynomial expressions?

Tip: When expanding polynomials, carefully distribute each term to ensure all terms are accounted for.