https://file.aiquickdraw.com/mathbot/file/1729243521904jmeqd8ub.jpg

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!

Expand and fully simplify (x + 5)(x + 4)^2

Learn how to expand and simplify the expression (x + 5)(x + 4)^2 with a step-by-step approach. This problem explores polynomial expansion using the binomial square identity and distributive property. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation