Final answer:
After substituting x²=-1 into the polynomial 2x⁴+5x³-x²+5x-3, we find the result is -12, not zero. Thus, x²+1 is not a factor of the polynomial.
Step-by-step explanation:
To determine if x²+1 is a factor of 2x⁴+5x³-x²+5x-3, we would typically perform polynomial division or use synthetic division. However, for quick verification, a factor implies that the polynomial would equate to zero when the factor is set to zero. If x²+1 is indeed a factor, substituting x²=-1 into the polynomial should result in zero.
Substituting -1 for x² in the polynomial gives us 2(-1)⁴ + 5(-1)³ - (-1) + 5(-1) - 3. Simplifying this, we have 2(1) - 5 - 1 - 5 - 3. Calculating further, we get 2 - 5 - 1 - 5 - 3 = -12, which is not zero. Therefore, x²+1 is not a factor of the given polynomial because the substitution does not result in zero.