Question

How would you solve ((-x^2)-(12x^3)-8)^2

Answer 1

Answer:

$\left(\left(-x^2\right)-\left(12x^3\right)-8\right)^2:\quad 144x^6+24x^5+x^4+192x^3+16x^2+64$

Step-by-step explanation:

Considering the expression

$\left(\left(-x^2\right)-\left(12x^3\right)-8\right)^2$

Solving the expression

$\left(\left(-x^2\right)-\left(12x^3\right)-8\right)^2$

$\mathrm{Remove\:parentheses}:\quad \left(a\right)=a$

$\left(-x^2-12x^3-8\right)^2$

As

$\left(-x^2-12x^3-8\right)^2=\left(-x^2-12x^3-8\right)\left(-x^2-12x^3-8\right)$

So

$\left(-12x^3-x^2-8\right)\left(-12x^3-x^2-8\right)$

$\mathrm{Expand}\:\left(-x^2-12x^3-8\right)\left(-x^2-12x^3-8\right):\quad 24x^5+144x^6+192x^3+x^4+16x^2+64$

i.e.

$\left(-x^2\right)\left(-x^2\right)+\left(-x^2\right)\left(-12x^3\right)+\left(-x^2\right)\left(-8\right)+\left(-12x^3\right)\left(-x^2\right)+\left(-12x^3\right)\left(-12x^3\right)+\left(-12x^3\right)\left(-8\right)+\left(-8\right)\left(-x^2\right)+\left(-8\right)\left(-12x^3\right)+\left(-8\right)\left(-8\right)$

$\mathrm{Apply\:minus-plus\:rules}$

$\left(-a\right)\left(-b\right)=ab$

$x^2x^2+12x^3x^2+8x^2+12x^3x^2+12\cdot \:12x^3x^3+12\cdot \:8x^3+8x^2+8\cdot \:12x^3+8\cdot \:8$

$\mathrm{Group\:like\:terms}$

$12x^3x^2+12x^3x^2+12\cdot \:12x^3x^3+12\cdot \:8x^3+8\cdot \:12x^3+x^2x^2+8x^2+8x^2+8\cdot \:8$

$\mathrm{Add\:similar\:elements:}\:8x^2+8x^2=16x^2$

$12x^3x^2+12x^3x^2+12\cdot \:12x^3x^3+12\cdot \:8x^3+8\cdot \:12x^3+x^2x^2+16x^2+8\cdot \:8$

$\mathrm{Add\:similar\:elements:}\:12x^3x^2+12x^3x^2=24x^3x^2$

$24x^3x^2+12\cdot \:12x^3x^3+12\cdot \:8x^3+8\cdot \:12x^3+x^2x^2+16x^2+8\cdot \:8$

$\mathrm{Add\:similar\:elements:}\:12\cdot \:8x^3+8\cdot \:12x^3=2\cdot \:8\cdot \:12x^3$

$24x^3x^2+12\cdot \:12x^3x^3+2\cdot \:8\cdot \:12x^3+x^2x^2+16x^2+8\cdot \:8$

As

$24x^3x^2=24x^5$

$12\cdot \:12x^3x^3=144x^6$

$2\cdot \:8\cdot \:12x^3=192x^3$

$x^2x^2=x^4$

$8\cdot \:8=64$

So

$24x^5+144x^6+192x^3+x^4+16x^2+64$

$\mathrm{Rewrite\:in\:standard\:form}$

$144x^6+24x^5+x^4+192x^3+16x^2+64$

Therefore,

$\left(\left(-x^2\right)-\left(12x^3\right)-8\right)^2:\quad 144x^6+24x^5+x^4+192x^3+16x^2+64$

Keywords: expression

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