Question

$$(x^2 - 2x - 4)(x^2 - 3x - 5)$$

Explanation:

Step1: Expand the product

We use the distributive property (FOIL method for polynomials) to multiply \((x^2 - 2x - 4)(x^2 - 3x - 5)\). First, multiply \(x^2\) by each term in the second polynomial: \(x^2 \cdot x^2 = x^4\), \(x^2 \cdot (-3x)= -3x^3\), \(x^2 \cdot (-5)= -5x^2\). Then, multiply \(-2x\) by each term: \(-2x \cdot x^2 = -2x^3\), \(-2x \cdot (-3x)= 6x^2\), \(-2x \cdot (-5)= 10x\). Next, multiply \(-4\) by each term: \(-4 \cdot x^2 = -4x^2\), \(-4 \cdot (-3x)= 12x\), \(-4 \cdot (-5)= 20\).

Step2: Combine like terms

Now, combine the like - degree terms:

• For \(x^4\): We have only \(x^4\).
• For \(x^3\): \(-3x^3-2x^3=-5x^3\)
• For \(x^2\): \(-5x^2 + 6x^2-4x^2=-3x^2\)
• For \(x\): \(10x + 12x = 22x\)
• For the constant term: We have \(20\)

Putting it all together, the expanded form is \(x^4-5x^3 - 3x^2+22x + 20\)

Answer:

The expanded form of \((x^2 - 2x - 4)(x^2 - 3x - 5)\) is \(x^4-5x^3-3x^2 + 22x+20\)