Question

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

Answer 1

To expand the expression \((x^2 - 3x + 1)(x^2 + 4x + 4)\), we will use the distributive property (also known as the FOIL method for binomials) for each term in the first polynomial multiplied by each term in the second polynomial:

1. **First, distribute \(x^2\)**:
\[
x^2 \cdot (x^2 + 4x + 4) = x^4 + 4x^3 + 4x^2
\]

2. **Next, distribute \(-3x\)**:
\[
-3x \cdot (x^2 + 4x + 4) = -3x^3 - 12x^2 - 12x
\]

3. **Finally, distribute \(1\)**:
\[
1 \cdot (x^2 + 4x + 4) = x^2 + 4x + 4
\]

Now, we combine all of these results together:
\[
x^4 + 4x^3 + 4x^2 - 3x^3 - 12x^2 - 12x + x^2 + 4x + 4
\]

Next, we simplify by combining like terms:

- For \(x^4\):
\[
x^4 \quad (\text{only one term})
\]

- For \(x^3\):
\[
4x^3 - 3x^3 = x^3
\]

- For \(x^2\):
\[
4x^2 - 12x^2 + x^2 = -7x^2
\]

- For \(x\):
\[
-12x + 4x = -8x
\]

- For the constant term:
\[
4 \quad (\text{only one term})
\]

Putting all these together, we get the expanded polynomial in standard form:
\[
\boxed{x^4 + x^3 - 7x^2 - 8x + 4}
\]