4.5.7 ** Let $\lambda>0$, find the largest constant $C=C(\lambda)$, such that for all non-negative real numbers $x, y$, we have $x^{2}+y^{2}+\lambda x y \geqslant C(x+y)^{2}$.
Analysis for $\lambda$ in two cases:
(1) When $\lambda \geqslant 2$, $x^{2}+y^{2}+\lambda x y \geqslant x^{2}+y^{2}+2 x y=(x+y)^{2}$, equality holds in the above inequality when $x y=0$.
(2) When $0<\lambda<2$,
$$
\begin{aligned}
x^{2}+y^{2}+\lambda x y & -(x+y)^{2}-(2-\lambda) x y \\
& \geqslant(x+y)^{2}-(2-\lambda)\left(\frac{x+y}{2}\right)^{2}=\frac{2+\lambda}{4}(x+y)^{2},
\end{aligned}
$$

equality holds when $x=y$.
Therefore, the largest constant $C(\lambda)=\left\{\begin{array}{ll}1, & \text { when } \lambda \geqslant 2 \text {, } \\ \frac{2+\lambda}{4}, & \text { when } 0<\lambda<2 \text { . }\end{array}\right.$