(3x+x^2)^2-x^2(x-5)(x+5)+2x(8-3x^2)

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression: (3x+x^2)^2-x^2(x-5)(x+5)+2x(8-3x^2)

This article will guide you through simplifying the algebraic expression: (3x+x^2)^2-x^2(x-5)(x+5)+2x(8-3x^2). We will break down each step to understand the process.

Expanding the Expression

1. Expanding the first term:

   • (3x+x^2)^2 = (3x+x^2)(3x+x^2)
   • Using the FOIL method (First, Outer, Inner, Last):
     • First: 3x * 3x = 9x^2
     • Outer: 3x * x^2 = 3x^3
     • Inner: x^2 * 3x = 3x^3
     • Last: x^2 * x^2 = x^4
   • Combining the terms: 9x^2 + 6x^3 + x^4

2. Expanding the second term:

   • x^2(x-5)(x+5)
   • Recognizing the difference of squares pattern: (a-b)(a+b) = a^2 - b^2
   • Applying the pattern: x^2(x^2 - 25)
   • Expanding: x^4 - 25x^2

3. Expanding the third term:

   • 2x(8-3x^2)
   • Distributing: 16x - 6x^3

Combining Terms

Now, we have: 9x^2 + 6x^3 + x^4 - (x^4 - 25x^2) + 16x - 6x^3

• Combining like terms:
  • x^4 - x^4 = 0
  • 6x^3 - 6x^3 = 0
  • 9x^2 + 25x^2 = 34x^2

This leaves us with: 34x^2 + 16x

Final Simplified Expression

Therefore, the simplified form of the given expression is: 34x^2 + 16x