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\(12y-9x^2+36-3x^2y\)
\(=\left(36-9x^2\right)-\left(3x^2y-12y\right)\)
\(=-9\left(x^2-4\right)-3y\left(x^2-4\right)\)
\(=\left(-3\right)\left(x-2\right)\left(x+2\right)\left(3+y\right)\)
\(=x^2+2x\cdot\frac{1}{2}+\frac{1}{4}-\left(\frac{\sqrt{23}}{2}i\right)^2\)
\(=\left(x+\frac{1}{2}\right)^2\)\(-\left(\frac{\sqrt{23}}{2}i\right)^2\)
\(\left(x+\frac{1}{2}-\frac{\sqrt{23}}{2}i\right)\left(x+\frac{1}{2}+\frac{\sqrt[]{23}}{2}i\right)\)
\(x^2+7x+12=x\left(x+3\right)+4\left(x+3\right)=\left(x+3\right)\left(x+4\right)\)
\(=x^2+3x+4x+12\)
\(=x\left(x+3\right)+4\left(x+3\right)\)
\(=\left(x+3\right)\left(x+4\right)\)
\(\dfrac{1}{4}x^2+2xy+4y^2=\left(\dfrac{1}{2}x+2y\right)^2\)
\(x^4-5x^2y^2+4y^4\)
\(=\left(x^2\right)^2-2x^22y^2+\left(2y^2\right)^2-x^2y^2\)
\(=\left(x^2-2y^2\right)^2-\left(xy\right)^2\)
\(=\left(x^2-2y^2-xy\right)\left(x^2-2y^2+xy\right)\)