What's the intuition behind nonlinear predictors in multivariate regression?
Not quite sure if this is what your question actually is. However, when you have a data generating process such as $y = 3 + 2x$ (with some zero-mean &quot;gaussian noise&quot;), your model could look like:
$$ y = \beta_0 + \beta_1 x + u,$$
and you will find the $\beta_0 = 3$ and $\beta_1 = 2$.
When you have a data generating process such as $y=3+2x+3x^2$ you need to do a &quot;basis expansion&quot; in your model to capture the additional complexity of the function and you could write a model such as:
$$ y = \beta_0 + \beta_1 x + \beta_2 x^2+ u.$$
The second model will capture the non-linear (quadratic) DGP &quot;perfectly&quot;.
df = data.frame(x1=seq(-10,20,by=0.2))

df$v = df$x1^2 
df$err = rnorm(nrow(df), mean = 0, sd = 1)
df$y = df$v + df$err 

plot(df$x1,df$y)

reg1 = lm(y~x1,data=df)
summary(reg1)
lines(df$x1,predict(reg1,newdata=df),col=&quot;red&quot;)

reg2 = lm(y~poly(x1,2,raw=T),data=df)
summary(reg2)
lines(df$x1,predict(reg2,newdata=df),col=&quot;blue&quot;)