$$ 4 ( ( \frac { \sqrt { 3 } } { 2 } x - \frac { 1 } { 2 } y ) ^ { 2 } ) + ( 2 \sqrt { 3 } ( \frac { \sqrt { 3 } } { 2 } x - \frac { 1 } { 2 } y ) ( \frac { 1 } { 2 } x + \frac { \sqrt { 3 } } { 2 } y ) ) + 2 ( ( \frac { 1 } { 2 } x + \frac { \sqrt { 3 } } { 2 } y ) ^ { 2 } ) + 10 \sqrt { 3 } ( \frac { \sqrt { 3 } } { 2 } x - \frac { 1 } { 2 } y ) + 10 ( \frac { 1 } { 2 } x + \frac { \sqrt { 3 } } { 2 } y ) = 5 $$

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UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Simplify:
  $$4\left(\left(\frac{\sqrt{3}}{2}x-\frac{1}{2}y\right)^{2}\right)+\left(2\sqrt{3}\times \left(\frac{\sqrt{3}}{2}x-\frac{1}{2}y\right)\left(\frac{1}{2}x+\frac{\sqrt{3}}{2}y\right)\right)+2\left(\left(\frac{1}{2}x+\frac{\sqrt{3}}{2}y\right)^{2}\right)+10\sqrt{3}\times \left(\frac{\sqrt{3}}{2}x-\frac{1}{2}y\right)+10\left(\frac{1}{2}x+\frac{\sqrt{3}}{2}y\right)$$
- step1: Calculate:
  $$4\left(\frac{\sqrt{3}}{2}x-\frac{1}{2}y\right)^{2}+\left(2\sqrt{3}\times \left(\frac{\sqrt{3}}{2}x-\frac{1}{2}y\right)\left(\frac{1}{2}x+\frac{\sqrt{3}}{2}y\right)\right)+2\left(\left(\frac{1}{2}x+\frac{\sqrt{3}}{2}y\right)^{2}\right)+10\sqrt{3}\times \left(\frac{\sqrt{3}}{2}x-\frac{1}{2}y\right)+10\left(\frac{1}{2}x+\frac{\sqrt{3}}{2}y\right)$$
- step2: Calculate:
  $$4\left(\frac{\sqrt{3}}{2}x-\frac{1}{2}y\right)^{2}+\left(2\sqrt{3}\times \left(\frac{\sqrt{3}}{2}x-\frac{1}{2}y\right)\left(\frac{1}{2}x+\frac{\sqrt{3}}{2}y\right)\right)+2\left(\frac{1}{2}x+\frac{\sqrt{3}}{2}y\right)^{2}+10\sqrt{3}\times \left(\frac{\sqrt{3}}{2}x-\frac{1}{2}y\right)+10\left(\frac{1}{2}x+\frac{\sqrt{3}}{2}y\right)$$
- step3: Calculate:
  $$4\left(\frac{\sqrt{3}}{2}x-\frac{1}{2}y\right)^{2}+2\sqrt{3}\times \left(\frac{\sqrt{3}}{2}x-\frac{1}{2}y\right)\left(\frac{1}{2}x+\frac{\sqrt{3}}{2}y\right)+2\left(\frac{1}{2}x+\frac{\sqrt{3}}{2}y\right)^{2}+10\sqrt{3}\times \left(\frac{\sqrt{3}}{2}x-\frac{1}{2}y\right)+10\left(\frac{1}{2}x+\frac{\sqrt{3}}{2}y\right)$$
- step4: Expand the expression:
  $$3x^{2}-2\sqrt{3}\times xy+y^{2}+2\sqrt{3}\times \left(\frac{\sqrt{3}}{2}x-\frac{1}{2}y\right)\left(\frac{1}{2}x+\frac{\sqrt{3}}{2}y\right)+2\left(\frac{1}{2}x+\frac{\sqrt{3}}{2}y\right)^{2}+10\sqrt{3}\times \left(\frac{\sqrt{3}}{2}x-\frac{1}{2}y\right)+10\left(\frac{1}{2}x+\frac{\sqrt{3}}{2}y\right)$$
- step5: Expand the expression:
  $$3x^{2}-2\sqrt{3}\times xy+y^{2}+\frac{3}{2}x^{2}+\sqrt{3}\times xy-\frac{3}{2}y^{2}+2\left(\frac{1}{2}x+\frac{\sqrt{3}}{2}y\right)^{2}+10\sqrt{3}\times \left(\frac{\sqrt{3}}{2}x-\frac{1}{2}y\right)+10\left(\frac{1}{2}x+\frac{\sqrt{3}}{2}y\right)$$
- step6: Expand the expression:
  $$3x^{2}-2\sqrt{3}\times xy+y^{2}+\frac{3}{2}x^{2}+\sqrt{3}\times xy-\frac{3}{2}y^{2}+\frac{1}{2}x^{2}+\sqrt{3}\times xy+\frac{3}{2}y^{2}+10\sqrt{3}\times \left(\frac{\sqrt{3}}{2}x-\frac{1}{2}y\right)+10\left(\frac{1}{2}x+\frac{\sqrt{3}}{2}y\right)$$
- step7: Expand the expression:
  $$3x^{2}-2\sqrt{3}\times xy+y^{2}+\frac{3}{2}x^{2}+\sqrt{3}\times xy-\frac{3}{2}y^{2}+\frac{1}{2}x^{2}+\sqrt{3}\times xy+\frac{3}{2}y^{2}+15x-5\sqrt{3}\times y+10\left(\frac{1}{2}x+\frac{\sqrt{3}}{2}y\right)$$
- step8: Expand the expression:
  $$3x^{2}-2\sqrt{3}\times xy+y^{2}+\frac{3}{2}x^{2}+\sqrt{3}\times xy-\frac{3}{2}y^{2}+\frac{1}{2}x^{2}+\sqrt{3}\times xy+\frac{3}{2}y^{2}+15x-5\sqrt{3}\times y+5x+5\sqrt{3}\times y$$
- step9: Add the terms:
  $$5x^{2}+0+y^{2}-\frac{3}{2}y^{2}+\frac{3}{2}y^{2}+15x-5\sqrt{3}\times y+5x+5\sqrt{3}\times y$$
- step10: Remove 0:
  $$5x^{2}+y^{2}-\frac{3}{2}y^{2}+\frac{3}{2}y^{2}+15x-5\sqrt{3}\times y+5x+5\sqrt{3}\times y$$
- step11: Calculate:
  $$5x^{2}+y^{2}+20x$$
Let's simplify the given expression step by step.

The original expression is:

$$
4 \left( \left( \frac{\sqrt{3}}{2} x - \frac{1}{2} y \right)^{2} \right) + \left( 2 \sqrt{3} \left( \frac{\sqrt{3}}{2} x - \frac{1}{2} y \right) \left( \frac{1}{2} x + \frac{\sqrt{3}}{2} y \right) \right) + 2 \left( \left( \frac{1}{2} x + \frac{\sqrt{3}}{2} y \right)^{2} \right) + 10 \sqrt{3} \left( \frac{\sqrt{3}}{2} x - \frac{1}{2} y \right) + 10 \left( \frac{1}{2} x + \frac{\sqrt{3}}{2} y \right) = 5
$$

After expanding and simplifying the expression, we find that it simplifies to:

$$
5x^{2} + y^{2} + 20x
$$

Now, we can set this equal to 5:

$$
5x^{2} + y^{2} + 20x = 5
$$

To solve for $$x$$ and $$y$$, we can rearrange the equation:

$$
5x^{2} + y^{2} + 20x - 5 = 0
$$

This is a quadratic equation in terms of $$x$$ and $$y$$. If you need to find specific values for $$x$$ and $$y$$, please provide additional conditions or constraints.
The simplified equation is $$5x^{2} + y^{2} + 20x = 5$$.

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