Question

PLATO; can somebody please help me with the following question below:

Answer 1

$\bf \begin{cases} f(x) = &x-6\\ g(x) = &x^{\frac{1}{2}}(x+3)\\ &\sqrt{x}(x+3)\\ &x\sqrt{x}+3\sqrt{x} \end{cases} \\\\\\ f(x) \times g(x)\implies \begin{array}{rllll} x\sqrt{x}+3\sqrt{x}\\\\ \times x\\\cline{1-1} x^2\sqrt{x}+3x\sqrt{x} \end{array}\qquad +\qquad \begin{array}{rllll} x\sqrt{x}+3\sqrt{x}\\\\ \times -6\\\cline{1-1} -6x\sqrt{x}-18\sqrt{x} \end{array}$

$\bf x^2\sqrt{x}\stackrel{like-terms}{+3x\sqrt{x}-6x\sqrt{x}}-18\sqrt{x}\implies x^2\sqrt{x}-3x\sqrt{x}-18\sqrt{x} \\\\\\ \sqrt{(x^2)^2 x}-3\sqrt{x^2 x}-18\sqrt{x}\implies \sqrt{x^4 x}-3\sqrt{x^2 x}-18\sqrt{x} \\\\\\ \sqrt{x^{4+1}}-3\sqrt{x^{2+1}}-18\sqrt{x}\implies \boxed{\sqrt{x^5}-3\sqrt{x^3}-18\sqrt{x}}$