Question

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

Answer 1

To expand the expression \((4x^2 + x + 5)(x^2 - x + 1)\), we can apply the distributive property (also known as the FOIL method for binomials). We'll multiply each term in the first polynomial by each term in the second polynomial.

1. Multiply \(4x^2\) by each term in \((x^2 - x + 1)\):
- \(4x^2 \cdot x^2 = 4x^4\)
- \(4x^2 \cdot (-x) = -4x^3\)
- \(4x^2 \cdot 1 = 4x^2\)

2. Multiply \(x\) by each term in \((x^2 - x + 1)\):
- \(x \cdot x^2 = x^3\)
- \(x \cdot (-x) = -x^2\)
- \(x \cdot 1 = x\)

3. Multiply \(5\) by each term in \((x^2 - x + 1)\):
- \(5 \cdot x^2 = 5x^2\)
- \(5 \cdot (-x) = -5x\)
- \(5 \cdot 1 = 5\)

Now, we can combine all the terms obtained from the multiplications:

### Gather the terms:
- From \(4x^2\):
- \(4x^4 - 4x^3 + 4x^2\)

- From \(x\):
- \(x^3 - x^2 + x\)

- From \(5\):
- \(5x^2 - 5x + 5\)

### Combine like terms:
- Overall expression:
\(4x^4\)
- Combine the \(x^3\) terms:
\(-4x^3 + x^3 = -3x^3\)
- Combine the \(x^2\) terms:
\(4x^2 - x^2 + 5x^2 = 8x^2\)
- Combine the \(x\) terms:
\(x - 5x = -4x\)
- The constant term is \(5\).

### Final Combined Expression:
Putting it all together, we get:
\[
4x^4 - 3x^3 + 8x^2 - 4x + 5
\]

Thus, the expanded expression in standard form is:
\[
\boxed{4x^4 - 3x^3 + 8x^2 - 4x + 5}
\]