To expand and simplify the expression $(x + 5)(x + 4)^2$, let's proceed step by step:

Step 1: Expand $(x + 4)^2$

Using the identity $(a + b)^2 = a^2 + 2ab + b^2$, we can expand $(x + 4)^2$: $(x + 4)^2 = x^2 + 8x + 16$

Step 2: Multiply $(x + 5)$ by the expanded form of $(x + 4)^2$

Now, distribute $(x + 5)$ over the expanded form: $(x + 5)(x^2 + 8x + 16)$ Distribute each term of $(x + 5)$ across $(x^2 + 8x + 16)$:

• $x \cdot (x^2 + 8x + 16) = x^3 + 8x^2 + 16x$
• $5 \cdot (x^2 + 8x + 16) = 5x^2 + 40x + 80$

Step 3: Combine like terms

Now, add the two results from the distribution: $x^3 + 8x^2 + 16x + 5x^2 + 40x + 80$ Combine like terms: $x^3 + (8x^2 + 5x^2) + (16x + 40x) + 80$ This simplifies to: $x^3 + 13x^2 + 56x + 80$

Thus, the expanded and simplified form of $(x + 5)(x + 4)^2$ is: $x^3 + 13x^2 + 56x + 80$

Would you like further explanation on any step?

Related Questions:

1. How would you approach simplifying $(x - 3)(x + 6)^2$?
2. What happens if you change the exponents in this expression to higher powers?
3. How does expanding $(x + 5)^3$ differ from expanding a quadratic expression?
4. Can you expand and simplify $(2x + 3)(x - 4)^2$?
5. How do the distributive property and FOIL method compare for expanding expressions?

Tip:

When expanding powers of binomials, always check if you can apply special identities like $(a + b)^2 = a^2 + 2ab + b^2$ to save time and reduce mistakes!