Factor 5(x^2+4)^4 (2x)(x-2)^4+(x^2+4)^5(4)(x-2)^3
Write it out
2(x^2+4)^4 (x-2)^3[ 5x(x-2) + 2(x^2+4) ]
= 2(x^2+4)^4 (x-2)^3 (7x^2 - 10x + 8)
To factor the expression 5(x^2+4)^4 (2x)(x-2)^4+(x^2+4)^5(4)(x-2)^3, we can first observe that both terms have a common factor of (x^2+4)^4. So let's factor it out:
= (x^2+4)^4 [5(2x)(x-2)^4 + (x^2+4)(4)(x-2)^3]
Next, let's simplify both terms within the square brackets:
= (x^2+4)^4 [10x(x-2)^4 + 4(x^2+4)(x-2)^3]
Now, let's focus on the second term within the square brackets and further simplify it:
= (x^2+4)^4 [10x(x-2)^4 + 4(x^2+4)(x-2)^3]
= (x^2+4)^4 [10x(x-2)^4 + 4(x^5-8x^4+12x^3-32x^2+16x-32)]
= (x^2+4)^4 [10x(x-2)^4 + 4x^5-32x^4+48x^3-128x^2+64x-128]
Now, we can distribute the common factors to both terms within the square brackets:
= (x^2+4)^4 [10x(x-2)^4] + (x^2+4)^4 [4x^5-32x^4+48x^3-128x^2+64x-128]
Since both terms still have a common factor of (x-2)^4, we can factor it out:
= (x^2+4)^4 (x-2)^4 [10x + (4x^5-32x^4+48x^3-128x^2+64x-128)]
Finally, we have factored the given expression as:
= (x^2+4)^4 (x-2)^4 [10x + 4x^5 - 32x^4 + 48x^3 - 128x^2 + 64x - 128]