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One proposed problem

A while ago I made a math problem, here I will attach it

Let  g  be a function such that \textrm{Let} \; g \; \textrm{be a function such that} g:R↦R,  g(x)=∑i=1neixi  where  ∀i∈[1..n](ei∈r) \newcommand{\twodots}{\mathinner {\ldotp \ldotp}} g:\mathbb{R} \mapsto \mathbb{R},\; g(x)= \sum_{i=1}^{n} e_ix^i \; \textrm{where}\; \forall i \in [1\twodots n](e_i\in\mathbb{r})

Let  f  be a function such that   \textrm{Let}\; f \; \textrm{be a function such that}\; f:R5↦R,  f(a,b,c,d,x)=g(x−a)⋅d+g(c−x)⋅bg(c−a) f:\mathbb{R}^5 \mapsto \mathbb{R},\; f(a,b,c,d,x)=\frac{g(x-a)\cdot d+g(c-x)\cdot b}{g(c-a)}

Determine the truthfulness of the proposition \textrm{Determine the truthfulness of the proposition}

∀a,b,c,d∈R,a≠c(∃p,q∈R(f(a,b,c,d,p)=b∧f(a,b,c,d,q)=d)) \forall a,b,c,d \in \mathbb{R},a \neq c (\exists p,q \in \mathbb{R}(f(a,b,c,d,p)=b \wedge f(a,b,c,d,q)=d))

Note: every  ei  is constant amongts every instance of  g \textrm{Note: every}\; e_i \; \textrm{is constant amongts every instance of}\; g