$$ f(x)=x_1^3 \, \cdot \, x_2$$
$$p_1=10\quad p_2 = 3 \quad Budget = 150 $$
$$x_1 \cdot p_1 +x_2 \cdot p_2 =150$$
$$x_2 \cdot p_2 =150-x_1 \cdot p_1 $$
$$x_2 =\frac{150-x_1 \cdot p_1 }{ p_2}$$
$$ f(x)=x_1^3 \, \cdot \, \frac{150-x_1 \cdot p_1 }{ p_2}$$
$$ f(x)= \, \frac{150\, x_1^3-\, x_1^4 \cdot p_1 }{ p_2}$$
$$ f'(x)= \, \frac{450\, x_1^2-4\, x_1^3 \cdot p_1 }{ p_2}$$
$$ f'(x)=0$$
$$ 0= \, \frac{450\, x_1^2-4\, x_1^3 \cdot p_1 }{ p_2}$$
$$ 0= x^2_1\, \cdot \, \frac{450\, -4\, x_1 \cdot p_1 }{ p_2}$$
$$ 0= 450\, -4\, x_1 \cdot p_1 $$
$$ 4\, x_1 \cdot p_1= 450\, $$
$$ x_1 = \frac{450}{4\, p_1} $$
