Question

How do you simplify `(x-5)(x+3)(x-3)-[4(x^2-4x+1)]^2`?

Answer 1

`−16 x^4 +x^3 +201x^2 - 9x + 29`

Explanation:

Simplify
`(x−5)(x+3)(x−3)−[4(x^2−4x+1)]^2`

I don't know if you'd call it "simplifying," but you can clear those parentheses.

1) Multiply `(x +3)(x-3)` by recognizing the binomials as the factors of a Difference of Two Squares
After you have multiplied the binomials by memorization, you get this:
`(x−5)(x^2 - 9)−[4(x^2−4x+1)]^2`

2) Clear the parentheses inside the brackets by distributing the `4`
`(x−5)(x^2 - 9)−[4x^2−16x+4]^2`

3) Clear the brackets by raising all the powers inside by the power of `2` outside.
To raise a power to a power, multiply the exponents.
`(x−5)(x^2 - 9)−[4^2 x^4−16^2 x^2+4^2]`

4) Evaluate the constants
`(x−5)(x^2 - 9)−[16  x^4−196 x^2+16]`

5) Clear the brackets by distributing the minus sign
`(x−5)(x^2 - 9)−16 x^4+196 x^2-16`

6) Clear the parentheses by multiplying the binomials
`x^3 +5x^2 - 9x + 45−16 x^4+196 x^2-16`

7) Group like terms
`−16 x^4 +x^3    (+5x^2 +196 x^2)   - 9x     (+ 45-16)`

8) Combine like terms
`−16 x^4 +x^3 +201x^2 - 9x + 29` `larr` answer