1. The problem is to evaluate the expression $$\left(1024x^{10} - 23040x^{8} + 161280x^{6} - 403200x^{4} + 302400x^{2} - 30240\right)e^{-x^{2}}$$ at $x=2$. 2. We substitute $x=2$ into the polynomial part first: $$1024(2)^{10} - 23040(2)^{8} + 161280(2)^{6} - 403200(2)^{4} + 302400(2)^{2} - 30240$$ 3. Calculate each power: $2^{10} = 1024$ $2^{8} = 256$ $2^{6} = 64$ $2^{4} = 16$ $2^{2} = 4$ 4. Substitute these values: $$1024 \times 1024 - 23040 \times 256 + 161280 \times 64 - 403200 \times 16 + 302400 \times 4 - 30240$$ 5. Calculate each multiplication: $1024 \times 1024 = 1048576$ $23040 \times 256 = 5898240$ $161280 \times 64 = 10321920$ $403200 \times 16 = 6451200$ $302400 \times 4 = 1209600$ 6. Now substitute these back: $$1048576 - 5898240 + 10321920 - 6451200 + 1209600 - 30240$$ 7. Perform the additions and subtractions step by step: $1048576 - 5898240 = -4849664$ $-4849664 + 10321920 = 5472256$ $5472256 - 6451200 = -979944$ $-979944 + 1209600 = 229656$ $229656 - 30240 = 199416$ 8. Now calculate the exponential part: $$e^{-x^{2}} = e^{-(2)^{2}} = e^{-4}$$ 9. The final value is: $$199416 \times e^{-4}$$ This is the exact value of the expression at $x=2$. 10. If you want a decimal approximation, use $e^{-4} \approx 0.0183156$: $$199416 \times 0.0183156 \approx 3651.6$$ Hence, the value of the expression at $x=2$ is approximately $3651.6$.