Find the values of the constants in the following identity $2x^3+3x^2-14x-5=(ax+b)(x+3)(x+1)+C$ I'm working through identities but I can't figure out how to get further than multiplying out the above to get :
$$2x^3+3x^2-14x-5=2ax^3+3ax^2+3ax+bx^2+3bx+bx+3b+C$$
can someone give me a hint on what to do next?

By expanding and grouping like terms together, we obtain:
$$ \begin{align*}
2x^3+3x^2-14x-5&=(ax+b)(x+3)(x+1) +C\\
2x^3+3x^2-14x-5&=(ax+b)(x^2+4x+3) +C\\
2x^3+3x^2-14x-5&=(ax)(x^2+4x+3)+b(x^2+4x+3) +C\\
2x^3+3x^2-14x-5&=(ax^3+4ax^2+3ax)+(bx^2+4bx+3b) +C\\
2x^3+3x^2-14x-5&=(a)x^3+(4a+b)x^2+(3a+4b)x+(3b+C)
\end{align*} $$
Now let's compare coefficients. In particular, focus on the coefficients of $x^3$, $x^2$, and $x^0$ (the constant term). From this, we obtain:
$$ \begin{align*}
x^3 &: \boxed{2=a} \\
x^2 &: 3 = 4a+b \implies b = 3-4a=3-4(2) \implies \boxed{b=-5} \\
x^0 &: -5=3b+C \implies C=-5-3b=-5-3(-5) \implies \boxed{C=10}
\end{align*} $$
