Is this implication correct? Code Answer

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Let $a$ and $b$ two coprime integer numbers. Then there exist $u₁$ and $v₁$ such that $$au_1+bv_1=1tag1$$

Let us consider an equation of the form:
$$ah+bd=gtag2$$

where $h$ is an unknown multivariate integer polynomial and $d$ is known multivariate integer polynomial. From $(1)$ we get
$$au_1g+bv_1g=gtag3$$

Then by comparison of $(3)$ with $(2)$ we get
$$ah=au_1g$$
and

$$bd=bv_1g$$

that is,

$$h=u_1g$$
and

$$d=v_1g$$

My question is: Is this implication true? I cannot make a decision on that.

Answer

NO. For example if $ane 0ne b$ and $0ne h=-bd$ and $0ne d=ah.$ Then $$0=g=ah+bd=au_1g+bv_1g$$ but $$ahne 0=au_1g quad text {and }; bdne 0=bv_1g.$$

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