If $x+\frac{1}{x}=2,$ then find the value of $(x^4+\frac{4}{{\left(x\right)}^4})(x^3-\frac{4}{x^3}).$

The correct option/answer is -15. Note: While solving the equation $x+\frac{1}{x}=2$ gives $x=1$, substituting $x=1$ into the target expression yields:
$(x^4+\frac{4}{x^4})(x^3-\frac{4}{x^3})=(1+4)(1-4)=5\times (-3)=-15$.

Let us break down the solution step-by-step:

Step 1: Solve the given equation for x
We are given the equation:
$x+\frac{1}{x}=2$
Multiply both sides of the equation by $x$ to eliminate the fraction:
$x^2+1=2x$
Rearranging the terms gives a quadratic equation:
$x^2-2x+1=0$
This quadratic equation can be factored as a perfect square:
${(x-1)}^2=0$
Taking the square root of both sides, we find:
$x=1$

Step 2: Substitute x = 1 into the expression
Now we substitute $x=1$ into the expression we need to evaluate:
$(x^4+\frac{4}{x^4})(x^3-\frac{4}{x^3})$
Evaluating the first factor:
$1^4+\frac{4}{1^4}=1+4=5$
Evaluating the second factor:
$1^3-\frac{4}{1^3}=1-4=-3$
Multiply the two results:
$5\times (-3)=-15$

Therefore, the value of the expression is -15.