Question

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

Answer 1

To expand the expression \((x^2 - x - 5)(-x^2 + x + 2)\), we can use the distributive property (also known as the FOIL method for binomials).

We will distribute each term in the first polynomial to each term in the second polynomial.

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

Next, distribute \(-x\):

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

Now, distribute \(-5\):

7. \(-5 \cdot (-x^2) = 5x^2\)
8. \(-5 \cdot x = -5x\)
9. \(-5 \cdot 2 = -10\)

Now we will sum all these results:

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

Next, we combine like terms:

- For \(x^4\): \(-x^4\)
- For \(x^3\): \(x^3 + x^3 = 2x^3\)
- For \(x^2\): \(2x^2 - x^2 + 5x^2 = (2 - 1 + 5)x^2 = 6x^2\)
- For \(x\): \(-2x - 5x = -7x\)
- Constant term: \(-10\)

Putting it all together, we have:

\[
-x^4 + 2x^3 + 6x^2 - 7x - 10
\]

Thus, the expanded expression in standard form is:

\[
\boxed{-x^4 + 2x^3 + 6x^2 - 7x - 10}
\]