Transformation of a quadratic term into a multiplicative term

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Can we always transform any quadratic term into a "linear term $\times$ linear term + linear term" form? I mean transforming something like $a_1x_1^2+a_2x_2^2+a_3x_1x_2+a_4$ to $(b_1x_1+c_1)(b_2x_2+c_2)+d_1x_1+d_2x_2+d_3$.