(3x^2-2x+1)(x^2+2x+3)-4x(x^2-1)-3x^2(x^2+2)

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression: (3x^2-2x+1)(x^2+2x+3)-4x(x^2-1)-3x^2(x^2+2)

This article will guide you through the process of simplifying the given algebraic expression:

(3x^2-2x+1)(x^2+2x+3)-4x(x^2-1)-3x^2(x^2+2)

We will achieve this by using the distributive property and combining like terms.

Step 1: Expand the Products

• (3x^2-2x+1)(x^2+2x+3)
  • This is a product of two trinomials. We will use the distributive property (or FOIL method) to expand it.
  • 3x^2 * (x^2 + 2x + 3) = 3x^4 + 6x^3 + 9x^2
  • -2x * (x^2 + 2x + 3) = -2x^3 - 4x^2 - 6x
  • 1 * (x^2 + 2x + 3) = x^2 + 2x + 3
• -4x(x^2-1)
  • We distribute the -4x to both terms inside the parenthesis.
  • -4x * x^2 = -4x^3
  • -4x * -1 = 4x
• -3x^2(x^2+2)
  • Distribute -3x^2 to both terms inside the parenthesis.
  • -3x^2 * x^2 = -3x^4
  • -3x^2 * 2 = -6x^2

Step 2: Combine Like Terms

Now that we have expanded all the products, we need to combine terms with the same variable and exponent.

• x^4 terms: 3x^4 - 3x^4 = 0
• x^3 terms: 6x^3 - 2x^3 - 4x^3 = 0
• x^2 terms: 9x^2 - 4x^2 + x^2 - 6x^2 = 0
• x terms: -6x + 4x = -2x
• Constant term: 3

Final Result

After combining like terms, the simplified expression is:

-2x + 3

Therefore, the simplified form of the original expression is -2x + 3.