<s>[INST] If $20 x^2+13 x-15$ can be written as $(a x+b)(c x+d)$, where $a, b, c$ and $d$ are integers, what is $a+b+c+d$? 
If $20 x^2+13 x-15$ can be written as $(a x+b)(c x+d)$, where $a, b, c$ and $d$ are integers, what is $a+b+c+d$ ?

The quadratic formula tells me that $x=\dfrac{-13 \pm 20\sqrt{3}}{40}.$
How to proceed?
 [/INST] If the question is asking you to factor you need not necessarily solve for the roots.
Use the FOIL (First, Out, In, Last) method to expand $(ax + b)(cx + d)$ and simultaneously solve for the coefficients $a, b, c, d$.
We would have $$\begin{cases} ac = 20 \\ ad + bc = 13 \\ bd = -15 \end{cases}$$
Furthermore, once you become more familiar and proficient with FOIL, you can try to guess factors and evaluate cases to see which $2$ factors of $20$ and $2$ factors of $-15$ add up to $13$ directly. This would be as such:
$20 \to \{\pm1, \pm2, \pm4, \pm5, \pm10, \pm20\}$
$-15 \to \{\pm1, \pm3, \mp 5, \mp15\}$
See that $5 \cdot 5 + 4 \cdot -3 = 13$; this is exactly in the form $ad + bc$. Then, $5 \cdot 4 = 20$ and $5 \cdot -3 = -15$, as required.
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