Factor completely: \( 3 t^{2}+22 t+24 \)

To factor the quadratic expression \( 3t^2 + 22t + 24 \) completely, we start by looking for two numbers that multiply to the product of the coefficient of \( t^2 \) (which is 3) and the constant term (which is 24) — that is \( 3 \times 24 = 72 \), and add up to the coefficient of \( t \) (which is 22). The numbers 18 and 4 fit the bill because \( 18 \times 4 = 72 \) and \( 18 + 4 = 22 \). Next, we can rewrite the middle term: \[ 3t^2 + 18t + 4t + 24. \] Now we can group the terms: \[ (3t^2 + 18t) + (4t + 24). \] Factoring out common factors from each group gives us: \[ 3t(t + 6) + 4(t + 6). \] Now, we notice that \( (t + 6) \) is a common factor: \[ (t + 6)(3t + 4). \] Thus, the completely factored form of \( 3t^2 + 22t + 24 \) is \( (t + 6)(3t + 4) \). It's always a good idea to check your work by expanding back, just to ensure we haven't made any errors along the way!