Question

Expand the expression to a polynomial in standard form:
left parenthesis, x, squared, minus, x, plus, 4, right parenthesis, left parenthesis, 3, x, squared, plus, 3, x, minus, 1, right parenthesis
(x
2
−x+4)(3x
2
+3x−1)

Answer 1

To expand the expression \((x^2 - x + 4)(3x^2 + 3x - 1)\), we'll use the distributive property (also known as the FOIL method for binomials, though here we have a trinomial and a binomial).

Let's distribute each term in the first polynomial to each term in the second polynomial.

1. **Distributing \(x^2\)**:
- \(x^2 \cdot 3x^2 = 3x^4\)
- \(x^2 \cdot 3x = 3x^3\)
- \(x^2 \cdot (-1) = -x^2\)

2. **Distributing \(-x\)**:
- \(-x \cdot 3x^2 = -3x^3\)
- \(-x \cdot 3x = -3x^2\)
- \(-x \cdot (-1) = x\)

3. **Distributing \(4\)**:
- \(4 \cdot 3x^2 = 12x^2\)
- \(4 \cdot 3x = 12x\)
- \(4 \cdot (-1) = -4\)

Now, we’ll combine all of these terms together:

\[
3x^4 + 3x^3 - x^2 - 3x^3 - 3x^2 + x + 12x^2 + 12x - 4
\]

Next, we can combine like terms:

- **\(x^4\) terms**: \(3x^4\)
- **\(x^3\) terms**: \(3x^3 - 3x^3 = 0\)
- **\(x^2\) terms**: \(-x^2 - 3x^2 + 12x^2 = 8x^2\)
- **\(x\) terms**: \(x + 12x = 13x\)
- **Constant terms**: \(-4\)

Putting it all together, the expanded polynomial in standard form is:

\[
\boxed{3x^4 + 8x^2 + 13x - 4}
\]