Can You Factorize it?

Let $f(x) = 9x^4 + 9x^3 + 8x^2 + 9x + 9$ such that $\boxed{f(10) = 99899}$ .

Now we will factorize $\large f(x)$ as follows $f(x) = x^2 \left( 9x^2 + 9x + 8 + \dfrac{9}{x} + \dfrac{9}{x^2} \right)$

$= x^2 \left[ 9\left(x^2+\dfrac{1}{x^2} \right) + 9\left(x+\dfrac{1}{x} \right) + 8 \right]$ $Now~ set \left(x+\dfrac{1}{x} \right) = t \\ \implies x^2 + \dfrac{1}{x^2} = t^2 - 2 \hspace{1cm} \color{#20A900}{\text{(Squaring both sides)}} \\ \implies \text{Substituting in f(x) and factorising we get :} \\ \implies f(x) = x^2 \left[ \left(3t - 2 \right) \left(3t + 5 \right) \right] \\ \implies f(x) = x^2 \left[ 3\left(x+\dfrac{1}{x} \right) - 2 \right] \left[ 3\left(x + \dfrac{1}{x} \right) + 5 \right] \\ \implies f(x) = \left(3x^2 - 2x +3 \right) \left(3x^2 +5x +3 \right) \\ \implies f(x) = g(x)\cdot h(x) \hspace{1cm} \color{#20A900}{\text{(Say)}} \\ \implies Now ~99899 ~~~=f(10) \\ \hspace{6cm} = g(10)\cdot h(10) \\ \large \hspace{6cm} = \color{#69047E}{\boxed{283 \times 353}}$

$\large \text{Hence , } \color{#D61F06}{\boxed{a + b + c + d = 18}}$