Answer:
$$\left(x^{3}-6x^{1}\right)\frac{\mathrm{d}}{\mathrm{d}x}(2x^{2}+3)+\left(2x^{2}+3\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}-6x^{1})$$
$$\left(x^{3}-6x^{1}\right)\times 2\times 2x^{2-1}+\left(2x^{2}+3\right)\left(3x^{3-1}-6x^{1-1}\right)$$
$$\left(x^{3}-6x^{1}\right)\times 4x^{1}+\left(2x^{2}+3\right)\left(3x^{2}-6x^{0}\right)$$
$$x^{3}\times 4x^{1}-6x^{1}\times 4x^{1}+\left(2x^{2}+3\right)\left(3x^{2}-6x^{0}\right)$$
$$x^{3}\times 4x^{1}-6x^{1}\times 4x^{1}+2x^{2}\times 3x^{2}+2x^{2}\left(-6\right)x^{0}+3\times 3x^{2}+3\left(-6\right)x^{0}$$
$$4x^{3+1}+4\left(-6\right)x^{1+1}+3\times 2x^{2+2}-6\times 2x^{2}+3\times 3x^{2}-6\times 3x^{0}$$
$$4x^{4}-24x^{2}+6x^{4}-12x^{2}+9x^{2}-18x^{0}$$
$$10x^{4}-36x^{2}+9x^{2}-18x^{0}$$
$$10x^{4}-36x^{2}+9x^{2}-18$$