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Question

how do u simplify:

(2-3x)^5(3x^2-x^3)(5)(2-3x)^4(-3) all over (2-3x)^10

and

[(2x^2-1)2x+(x^2+3)4x]-(2x^-1)(x^2+3)2x all over (x^2+1)

Answer 1

ow!! that hurts my head...

Answer 2

(2-3x)^5(3x^2-x^3)(5)(2-3x)^4(-3) / (2-3x)^10

Let's space that out to be easier to read:
(2 - 3x)^5 * (3x^2 - x^3) * (5) * (2-3x)^4(-3) / (2-3x)^10

Step 1: Reorder.
(5)(-3) * (3x^2 - x^3) * (2 - 3x)^5 * (2-3x)^4 / (2-3x)^10

Step 2: Combine (2 - 3x)^5 * (2-3x)^4 / (2-3x)^10
(2 - 3x)^5 * (2-3x)^4 = (2 - 3x)^9
(2 - 3x)^9 / (2 - 3x)^10 = 1 / (2 - 3x)
Giving you:
-15(3x^2 - x^3) / (2 - 3x)
-15x^2(3 - x) / (2 - 3x)
15x^2(x - 3) / (2 - 3x) (solution!)

2.) [(2x^2-1)2x+(x^2+3)4x]-(2x^-1)(x^2 + 3)2x / (x^2+1)

Space it out:
[(2x^2 - 1)2x + (x^2 + 3) 4x] - (2x^-1)(x^2 + 3)2x / (x^2+1)

Simplify inside brackets:
[(2x^2 - 1)2x + (x^2 + 3) 4x] = 2x[(2x^2 - 1) + 2(x^2 + 3)]
2x[2x^2 - 1 + 2x^2 + 6] = 2x[4x^2 + 5]

Giving you:
2x[4x^2 + 5] - (2x^-1)(x^2 + 3)2x / (x^2+1)

Now, simplify behind the subtraction:
(2x^-1)(x^2 + 3)2x: Note that 2x^-1 = 2/x.
2 / x * (x^2 + 3)2x = 4(x^2 + 3)

Giving you:
2x[4x^2 + 5] - 4(x^2 + 3) / (x^2 + 1)

Now, expand the top part of the fraction out completely:
8x^3 - 4x^2 + 10x - 12 / (x^2 + 1)
2(4x^3 - 2x^2 + 5x - 6) / x^2 + 1

You can split this out so that you can create a mixed fraction:
2(4x^3 - 2x^2 + 4x - 2) / (x^2 + 1) + 2(x - 4) / (x^2 + 1)

The left side of this simplifies to: 2(4x^3 - 2x^2 + 4x - 2) / (x^2 + 1) = 2(4x^3 + 4x) - 2(2x^2 + 2) / (x^2 + 1) = 2(4x - 2)

Giving you the simplest answer:
2(4x - 2) + 2(x - 4)/(x^2 + 1)