If $(ax+1)(bx+3)=6x^2+cx+3$ for all values of $x$, and $a+b=5$, what are the two possible values for $c$ ?

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Expand the expression on the left side first:

$$(ax+1)(bx+3)$$ $$abx^2+bx+3ax+3$$

Let's compare this with the equation on the other side, taking care to identify corresponding parts:

$$\colorbox{aqua}{$abx^2$}+\colorbox{yellow}{$bx+3ax$}+\colorbox{chartreuse}{$3$} = \colorbox{aqua}{$6x^2$}+\colorbox{yellow}{$cx$}+\colorbox{chartreuse}{$3$}$$

From the above, we get the following equations:

We are given that $a+b=5$, which we can use along with $ab=6$ to identify possible values for $a$ and $b$. We can solve the system using tradional methods, or use a bit of mental math.

$$a \text{ and } b = 3 \text{ or } 2$$

$c$ can have two answers, based on the values we assign $a$ and $b$.

$$c=b+3a$$ $$c=\boxed{9 \text{ and } 11}$$