Question

Expand the expression to a polynomial in standard form:
left parenthesis, x, minus, 1, right parenthesis, left parenthesis, 2, x, plus, 1, right parenthesis, left parenthesis, x, minus, 6, right parenthesis, left parenthesis, x, plus, 1, right parenthesis
(x−1)(2x+1)(x−6)(x+1)

Answer 1

To expand the expression \((x - 1)(2x + 1)(x - 6)(x + 1)\), we can proceed step by step, expanding two factors at a time.

**Step 1: Expand \((x - 1)(2x + 1)\)**

\[
(x - 1)(2x + 1) = x(2x + 1) - 1(2x + 1)
\]
\[
= 2x^2 + x - 2x - 1
\]
\[
= 2x^2 - x - 1
\]

**Step 2: Expand \((x - 6)(x + 1)\)**

\[
(x - 6)(x + 1) = x(x + 1) - 6(x + 1)
\]
\[
= x^2 + x - 6x - 6
\]
\[
= x^2 - 5x - 6
\]

**Step 3: Now multiply the two results: \((2x^2 - x - 1)(x^2 - 5x - 6)\)**

We will use the distributive property (FOIL method) here:

\[
= 2x^2(x^2 - 5x - 6) - x(x^2 - 5x - 6) - 1(x^2 - 5x - 6)
\]

Calculating each component:

1. \(2x^2(x^2) = 2x^4\)
2. \(2x^2(-5x) = -10x^3\)
3. \(2x^2(-6) = -12x^2\)

Thus, we have:

\[
2x^4 - 10x^3 - 12x^2
\]

Next:

1. \(-x(x^2) = -x^3\)
2. \(-x(-5x) = 5x^2\)
3. \(-x(-6) = 6x\)

Thus, we have:

\[
-x^3 + 5x^2 + 6x
\]

Lastly:

1. \(-1(x^2) = -x^2\)
2. \(-1(-5x) = 5x\)
3. \(-1(-6) = 6\)

Thus, we have:

\[
-x^2 + 5x + 6
\]

**Combining all parts:**

\[
2x^4 - 10x^3 - 12x^2 - x^3 + 5x^2 + 6x - x^2 + 5x + 6
\]

Now we combine like terms:

- \(2x^4\)
- \(-10x^3 - x^3 = -11x^3\)
- \(-12x^2 + 5x^2 - x^2 = -8x^2\)
- \(6x + 5x = 11x\)
- Constant term is \(6\)

Putting it all together, we get:

\[
\boxed{2x^4 - 11x^3 - 8x^2 + 11x + 6}
\]