A garden is designed in a rectangular shape with dimensions represented by the quadratic functions of length and width. The length of the garden is given by \( L(x) = x^2 + 5x + 6 \), and the width is given by \( W(x) = 2x + 3 \). If the area of the garden is 120 square meters when \( x = 2 \), verify the dimensions of the garden and find the value of \( x \). Let's think step by step and output the final answer within \boxed{}.
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