Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial: 

(i) t² – 3, 2t⁴ + 3t³ – 2t² – 9t – 12 (ii) x² + 3x + 1, 3x⁴ + 5x³ – 7x² + 2x + 2 (iii) x³ – 3x + 1, x⁵ – 4x³ + x² + 3x + 1

(i)  Given $p\left(x\right)=2t^4+3t^3-2t^2-9t-12, g\left(x\right)=t^2-3.$

$$\begin{gathered} \begin{array}{ccccc}2t^2& +& 3t& +& 4\end{array} \\ \begin{array}{ccc}{t}^2& -& 3\end{array}\sqrt{\begin{array}{ccccc}2t^4& + 3t^3& - 2t^2& - 9t& - 12\\ 2t^4& + 0t^3& - 6t^2& & \\ (-) & & (+) & & \end{array}} \\ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ \begin{array}{cccc}3t^3& + 4t^2& - 9t& - 12\\ 3t^3& + 0t^2& - 9t& \\ (-) & & (+) & \end{array} \\ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ \begin{array}{ccc}4t^2& + ot& - 12\\ 4t^2& + 0t& - 12\\ (-) & & (+) \end{array} \\ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ 0 \end{gathered}$$

Here, the remainder is 0.

Therefore, g(x) is the factor of p(x).