Math Problem Statement

(x2+2)25(x2+2)2-3

Solution

The expression you've written is:

(x2+2)2[5(x2+2)23](2x)(x^2 + 2)^2 \left[ 5(x^2 + 2)^2 - 3 \right] (2x)

Step 1: Expand the expression

First, let's break down each part of this expression.

(x2+2)2(x^2 + 2)^2

This is a binomial squared, so we apply the formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2:

(x2+2)2=(x2)2+2(x2)(2)+22=x4+4x2+4(x^2 + 2)^2 = (x^2)^2 + 2(x^2)(2) + 2^2 = x^4 + 4x^2 + 4

5(x2+2)235(x^2 + 2)^2 - 3

Now, let's handle this part. We already know that (x2+2)2=x4+4x2+4(x^2 + 2)^2 = x^4 + 4x^2 + 4. So:

5(x2+2)23=5(x4+4x2+4)35(x^2 + 2)^2 - 3 = 5(x^4 + 4x^2 + 4) - 3

Distribute the 5:

=5x4+20x2+203=5x4+20x2+17= 5x^4 + 20x^2 + 20 - 3 = 5x^4 + 20x^2 + 17

Step 2: Substitute and simplify

Now, substitute these expanded terms back into the original expression:

(x4+4x2+4)(5x4+20x2+17)(2x)(x^4 + 4x^2 + 4) \left( 5x^4 + 20x^2 + 17 \right) (2x)

Distribute the terms

We multiply the two binomials first, and then multiply by 2x2x.

  1. Multiply (x4+4x2+4)(x^4 + 4x^2 + 4) by (5x4+20x2+17)(5x^4 + 20x^2 + 17).
  2. After that, multiply the resulting expression by 2x2x.

Would you like to continue with the detailed expansion, or would you like to ask a follow-up question? Let me know how you would like to proceed.


Related Questions:

  1. How can I simplify a polynomial expression involving exponents?
  2. What is the distributive property, and how do I apply it to polynomials?
  3. Can you explain how to expand binomials like (a+b)2(a + b)^2 and apply them to other terms?
  4. How do I factor expressions that contain powers of xx?
  5. What is the best way to organize multi-term polynomial expansions?

Tip: Always try to simplify each part of a complicated expression step by step to avoid missing terms or making algebraic errors.

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Math Problem Analysis

Mathematical Concepts

Algebra
Binomial Expansion
Polynomials
Distributive Property

Formulas

(a + b)^2 = a^2 + 2ab + b^2
a(b + c) = ab + ac

Theorems

Binomial Theorem
Distributive Property

Suitable Grade Level

Grades 9-11