Custom changes to polynomial function (polynom) for a variable in R

This question is more for my own curiosity. I was looking through the Polynom documenation pdf for R, and noticed several basic polynomial operations such as:

p <- poly.calc(1:5)
## -120 + 274*x - 225*x^2 + 85*x^3 - 15*x^4 + x^5

However, how would I represent 'x' if I had to solve an equation such as:

(x+1)3x^3 + (x+1)4x^2 + (x+1) 2x + 3 = 17

and as a bonus, what if I wanted to specify x is greater than 0, and would change the equation to (x+1)3x^3 + (x+1)4x^2 + (x+1) 2x + 3 ≈ 17 where x > 0? (if it is even possible)

I think you are asking something like the following:

With the library polynom:

library(polynom)
#(x+1)3x^3 + (x+1)4x^2 + (x+1) 2x + 3 - 17 = (x+1)(3x^3 + 4x^2 + 2x) - 14
p1 <- polynomial(coef = c(1, 1))
p2 <- polynomial(coef = c(0, 2, 4, 3))
p <- p1 * p2 - 14
roots <- solve(p)
real.roots <- Re(roots[Im(roots)==0])
# [1] -2.0554784  0.9069195
real.positive.roots <- real.roots[real.roots > 0]
# [1] 0.9069195
plot(p)
abline(h=0)
abline(v=real.roots, lty=2)
points(real.roots, rep(0, length(real.roots)))

With the function uniroot:

f <- function (x) (x+1)*3*x^3 + (x+1)*4*x^2 + (x+1)*2*x + 3 - 17
root1 <- uniroot(f, c(-2, 1), tol = 0.0001)$root
# [1] 0.9069388
root2 <- uniroot(f, c(-3, -2), tol = 0.0001)$root
# [1] -2.055487

x <- seq(-2.5,1.5,0.001)
plot(x, f(x), type='l')
abline(h=0)
abline(v=root1, lty=2)
abline(v=root2, lty=2)
points(root1, 0, col='red', pch=19)
points(root2, 0, col='red', pch=19)