The perfect square
Let's highlight the perfect square of the square three-member
$$v^{2} + \left(u^{2} + 2 u v\right)$$
Let us write down the identical expression
$$v^{2} + \left(u^{2} + 2 u v\right) = 0 v^{2} + \left(u^{2} + 2 u v + v^{2}\right)$$
or
$$v^{2} + \left(u^{2} + 2 u v\right) = 0 v^{2} + \left(u + v\right)^{2}$$
General simplification
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$$u^{2} + 2 u v + v^{2}$$
$$\left(u + v\right)^{2}$$
$$u^{2} + 2 u v + v^{2}$$
$$u^{2} + 2 u v + v^{2}$$
$$u^{2} + 2 u v + v^{2}$$
Combining rational expressions
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$$u \left(u + 2 v\right) + v^{2}$$
Rational denominator
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$$u^{2} + 2 u v + v^{2}$$
Assemble expression
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$$u^{2} + 2 u v + v^{2}$$