Question:

<p><strong>Step 1:</strong> Use the triangle angle sum property: $A + B + C = \pi$, so $A + B = \pi - C$.</p><p><strong>Step 2:</strong> Therefore $\tan(A+B) = \tan(\pi - C) = -\tan C$.</p><p><strong>Step 3:</strong> Using the addition formula: $\frac{\tan A + \tan B}{1 - \tan A \tan B} = -\tan C$.</p><p><strong>Step 4:</strong> Substitute $\tan B + \tan C = 5$, so $\tan B = 5 - \tan C$.</p><p><strong>Step 5:</strong> From the given $\tan A \tan C = 3$, we get $\tan A = \frac{3}{\tan C}$.</p><p><strong>Step 6:</strong> Substituting into the identity: $\frac{\frac{3}{\tan C} + 5 - \tan C}{1 - \frac{3}{\tan C}(5-\tan C)} = -\tan C$.</p><p><strong>Step 7:</strong> Solving yields two possible values of $\tan A$. The sum of all possible values is 10.</p><p><strong>Step 8:</strong> For both cases, all angles are acute, so the triangle is acute-angled.</p><p>∴ Answer is (a, c).</p>