Algebra and triangle

Since $a$ and $b$ are roots of $x^2 - (c+6)x+6(c+3) = 0$ , by Vieta's formula , we have

$\begin{cases} a + b = c + 6 & ...(1) \\ ab = 6(c+3) & ...(2) \end{cases}$

$\begin{aligned} (1)^2: \quad a^2 + 2ab + b^2 & = c^2 + 12c + 36 \\ a^2 + b^2 & = c^2 + 12c + 36 - 2\blue{ab} & \small \blue{\text{Note that }(2):\ ab = 6(c+3)} \\ & = c^2 + 12c + 36 - 2\blue{12c-36} \\ \implies a^2+b^2 & = c^2 \end{aligned}$

Therefore, the given triangle is a right triangle and the measure of the largest angle is $\boxed{90}^\circ$ .