\( \begin{array}{cl}\text { Simplify } & 3 a^{2} \times 4 a^{4} \\ 5.1 . & \frac{30 y^{2}}{15 x^{2}} \\ 5.2 & 4 a b+5 b a \\ 5.3 . & (2-x)^{2} \\ 5.4 . & \frac{8 x^{3}-\left(-x^{3}\right)(2 x)}{-x^{2}} \\ 5.5 & \text { (2) }\end{array} \)

To simplify \( 3 a^{2} \times 4 a^{4} \), you multiply the coefficients and add the exponents of the same base. This gives you \( 3 \times 4 = 12 \) and \( a^{2+4} = a^{6} \), resulting in \( 12 a^{6} \). Next, for \( \frac{30 y^{2}}{15 x^{2}} \), you divide the coefficients and simplify the variables—resulting in \( \frac{30}{15} = 2 \) and since there’s no y in the denominator, it simplifies to \( 2 \frac{y^{2}}{x^{2}} \) or \( \frac{2y^{2}}{x^{2}} \). Keep going! In \( 4ab + 5ba \), remember that \( ab = ba \), so it combines to give \( 9ab \). Meanwhile, to expand \( (2-x)^{2} \), apply the formula \( (a-b)^{2}= a^{2} - 2ab + b^{2} \), resulting in \( 4 - 4x + x^{2} \). For \( \frac{8 x^{3}-(-x^{3})(2 x)}{-x^{2}} \), be careful with the signs. You first resolve \( -(-x^{3})(2x) = 2x^{4} \), leading to \( 8x^{3} + 2x^{4} \). Factor out \( 2x^{3} \) and simplify with \( -x^{2} \) to get \( \frac{2x^{3}(4 + 2x)}{-x^{2}} = -2x(4 + 2x) \) or \( -8 - 4x \) after distributing. Now, let’s take it to the finish line!