By CS we have: \[\frac{(a^2+b^2+2c^2+3d^2)(6c^2+6d^2+3b^2+2a^2)}{(a+b)^2(c+d)^2}\geq \frac{(\sqrt{6}ac+\sqrt{6}bd+\sqrt{6}bc+\sqrt{6}ad)^2}{(a+b)^2(c+d)^2} = 6\]Equality occurs iff $\frac{a}{c\sqrt{6}} = \frac{b}{d\sqrt{6}} = \frac{c\sqrt{2}}{b\sqrt{3}} = \frac{d\sqrt{3}}{a\sqrt{2}}$ or $a:b:c:d = \frac{3^{\frac{3}{4}}}{2^{\frac{1}{4}}}:6^{\frac{1}{4}}:\frac{3^{\frac{1}{2}}}{2^{\frac{1}{2}}}:1$ (hopefully I haven't messed up the messy calculations), so there is an equality case and the minimum value is $6$.

AlperenINAN wrote: Let $a,b,c,d$ be positive real numbers. What is the minimum value of $$ \frac{(a^2+b^2+2c^2+3d^2)(2a^2+3b^2+6c^2+6d^2)}{(a+b)^2(c+d)^2}$$ More for this type of question: 1-) Let $a,b,c,d$ be positive real numbers. What is the minimum value of $\frac{(4a^2+3b^2+2c^2+24d^2)(12a^2+b^2+8c^2+6d^2)}{(a+b)^2(c+d)^2}$ 2-) Let $a,b,c,d$ be positive real numbers. What is the minimum value of $\frac{(4a^2+6b^2+12c^2+18d^2)(3a^2+2b^2+6c^2+9d^2)}{(a+b)^2(c+d)^2}$

Generalization 1 Let $a,b,c,d,k,m,n,p$ be positive reels, n be positive integer with x positive divisors($d_{1}<d_{2}<\cdots<d_{x}$). Then prove that $$\dfrac{\left(d_{k}a^2+d_{m}b^2+d_{n}c^2+d_{p}d^2\right)\left(d_{x+1-p}a^2+d_{x+1-n}b^2+d_{x+1-k}c^2+d_{x+1-m}d^2\right)}{(a+b)^2(c+d)^2}\geq n$$