the order of the factor ring $\mathbb{Z}_{15}\left [ x \right ] / \left \langle 3x^{2}+5x \right \rangle$ What is the order of the factor ring $\mathbb{Z}_{15}\left [ x \right ] / \left \langle 3x^{2}+5x \right \rangle$ ?
Since $x$ and $3x+5$ are not relatively prime polynomial in $\mathbb{Z}_{15}$, so I can't use the Chinese Remainder Theorem.
Any help please. Thank you.
Let $f=3x^{2}+5x$ and $I=\langle f \rangle$.


$I \ni 3f = 9x^2+15x = 9x^2$.
$I \ni 5f = 15x^2+25x = 25x = 10x = -5x$.
$I \ni 5xf - 3f = 10x^2 - 9x^2 = x^2$.


Thus, $I=\langle 5x, x^2 \rangle$.

Therefore, for every $g \in \mathbb{Z}_{15}[x]$ we have $g \equiv ax+b \bmod I$, with $a \in \{0,1,2,3,4\}$.

Thus, $\mathbb{Z}_{15}[x] / I$ has $5 \cdot 15 = 75$ elements.
Hint: Here, $3x^2+5x=x(3x+5)$ both factors are irreducible and relatively prime.
Use Chinese reminder theorem for rings to find an isomorphism and to  SO get the order of ring.