Expand the expression to a polynomial in standard form:
left parenthesis, 4, x, minus, 5, right parenthesis, left parenthesis, x, squared, plus, 4, x, plus, 1, right parenthesis
(4x−5)(x 
2
 +4x+1)

To expand the expression \( (4x - 5)(x^2 + 4x + 1) \), we'll apply the distributive property (also known as the FOIL method when dealing with binomials). This involves distributing each term in \( 4x - 5 \) to every term in \( x^2 + 4x + 1 \).

### Step-by-step solution:

\[
(4x - 5)(x^2 + 4x + 1)
\]

1. **Distribute** \( 4x \):
   \[
   4x \cdot (x^2 + 4x + 1) = 4x^3 + 16x^2 + 4x
   \]

2. **Distribute** \( -5 \):
   \[
   -5 \cdot (x^2 + 4x + 1) = -5x^2 - 20x - 5
   \]

3. **Combine like terms**:
   \[
   4x^3 + (16x^2 - 5x^2) + (4x - 20x) - 5
   \]

   This simplifies to:
   \[
   4x^3 + 11x^2 - 16x - 5
   \]

Thus, the expanded polynomial in standard form is:

\[
\boxed{4x^3 + 11x^2 - 16x - 5}
\]

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Would you like more details or have any questions on this? 

Here are 5 related questions to practice:
1. Expand \( (3x - 2)(x^2 + 5x + 3) \).
2. Expand and simplify \( (2x + 3)(x^2 - x + 4) \).
3. Factor \( 4x^3 + 11x^2 - 16x - 5 \) back into its factors.
4. How does the FOIL method apply to binomials?
5. What are the standard forms of polynomials?

**Tip**: When expanding, always combine like terms to simplify the expression!

Expand the expression to a polynomial in standard form: (4x−5)(x^2 + 4x + 1)

Expand and simplify the polynomial expression (4x−5)(x^2 + 4x + 1) using the distributive property. This problem involves key algebraic concepts such as polynomial expansion and the combination of like terms. Suitable for students in Grades 9-11.

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