Expand the expression to a polynomial in standard form:

Question

left parenthesis, x, squared, plus, x, minus, 2, right parenthesis, left parenthesis, x, squared, minus, x, minus, 6, right parenthesis
(x
2
+x−2)(x
2
−x−6)

Solve the problem just like the example below
⬇️
Expand the expression to a polynomial in standard form:
left parenthesis, x, squared, minus, x, minus, 2, right parenthesis, left parenthesis, minus, x, squared, plus, 3, x, minus, 6, right parenthesis
(x
2
−x−2)(−x
2
+3x−6)

minus, x, squared
−x
2
plus, 3, x
+3x
minus, 6
−6
x, squared
x
2
minus, x, to the power 4
−x
4
3, x, cubed
3x
3
minus, 6, x, squared
−6x
2
minus, x
−x
x, cubed
x
3
minus, 3, x, squared
−3x
2
6, x
6x
minus, 2
−2
2, x, squared
2x
2
minus, 6, x
−6x
12
12
Combine like terms:
Combine like terms:
minus, x, to the power 4 , plus, 4, x, cubed, minus, 7, x, squared, plus, 12
−x
4
+4x
3
−7x
2
+12

Answer 1

Sure! Here's the step-by-step expansion of the expression \((x^2 + x - 2)(x^2 - x - 6)\):

1. Write down the expression: \[ (x^2 + x - 2)(x^2 - x - 6) \]

2. Distribute each term in the first polynomial to each term in the second polynomial:

   • Distributing \(x^2\): \[ x^2 \cdot x^2 = x^4 \] \[ x^2 \cdot (-x) = -x^3 \] \[ x^2 \cdot (-6) = -6x^2 \]

   • Distributing \(x\): \[ x \cdot x^2 = x^3 \] \[ x \cdot (-x) = -x^2 \] \[ x \cdot (-6) = -6x \]

   • Distributing \(-2\): \[ -2 \cdot x^2 = -2x^2 \] \[ -2 \cdot (-x) = 2x \] \[ -2 \cdot (-6) = 12 \]

3. Combine all the results: \[ x^4 - x^3 - 6x^2 + x^3 - x^2 - 6x - 2x^2 + 2x + 12 \]

4. Combine like terms:

   • From the \(x^3\) terms: \(-x^3 + x^3 = 0\)
   • From the \(x^2\) terms: \(-6x^2 - x^2 - 2x^2 = -9x^2\)
   • From the \(x\) terms: \(-6x + 2x = -4x\)
   • The constant term is \(12\).

5. Write the final expression in standard form: \[ x^4 - 9x^2 - 4x + 12 \]

So the final expanded polynomial in standard form is: \[ \boxed{x^4 - 9x^2 - 4x + 12} \]