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(C) Let the new coordinates be $(x', y')$. The transformation equations are $x = x' + \frac{3}{2}$ and $y = y' - 2$.Substituting these into the given

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$4x^2+8xy+2y^2+9=0$

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(C) Let the new coordinates be $(x', y')$. The transformation equations are $x = x' + \frac{3}{2}$ and $y = y' - 2$.Substituting these into the given equation $2x^2 + 4xy + y^2 + 2x - 2y + 1 = 0$:$2(x' + \frac{3}{2})^2 + 4(x' + \frac{3}{2})(y' - 2) + (y' - 2)^2 + 2(x' + \frac{3}{2}) - 2(y' - 2) + 1 = 0$Expanding the terms:$2((x')^2 + 3x' + \frac{9}{4}) + 4(x'y' - 2x' + \frac{3}{2}y' - 3) + ((y')^2 - 4y' + 4) + 2x' + 3 - 2y' + 4 + 1 = 0$$2(x')^2 + 6x' + \frac{9}{2} + 4x'y' - 8x' + 6y' - 12 + (y')^2 - 4y' + 4 + 2x' - 2y' + 8 = 0$Combining like terms:$2(x')^2 + 4x'y' + (y')^2 + (6x' - 8x' + 2x') + (6y' - 4y' - 2y') + (\frac{9}{2} - 12 + 4 + 8) = 0$$2(x')^2 + 4x'y' + (y')^2 + \frac{9}{2} = 0$Multiplying by $2$ to clear the fraction:$4(x')^2 + 8x'y' + 2(y')^2 + 9 = 0$.Thus,the transformed equation is $4x^2 + 8xy + 2y^2 + 9 = 0$.

If the origin is shifted to the point $\left(\frac{3}{2},-2\right)$ by the translation of axes,then the transformed equation of $2x^2+4xy+y^2+2x-2y+1=0$ is

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