Expanding the Polynomial: (x^5 + 2x^4 + 3x^2 + x  3)(x^2 + 1)
This expression involves multiplying two polynomials: a fifthdegree polynomial (x^5 + 2x^4 + 3x^2 + x  3) and a seconddegree polynomial (x^2 + 1). To expand this expression, we'll use the distributive property (also known as FOIL for binomials).
StepbyStep Expansion

Distribute the first term of the first polynomial (x^5) over the second polynomial:
 x^5 * (x^2 + 1) = x^7 + x^5

Distribute the second term of the first polynomial (2x^4) over the second polynomial:
 2x^4 * (x^2 + 1) = 2x^6 + 2x^4

Distribute the third term of the first polynomial (3x^2) over the second polynomial:
 3x^2 * (x^2 + 1) = 3x^4 + 3x^2

Distribute the fourth term of the first polynomial (x) over the second polynomial:
 x * (x^2 + 1) = x^3 + x

Distribute the fifth term of the first polynomial (3) over the second polynomial:
 3 * (x^2 + 1) = 3x^2  3

Finally, combine all the terms:
 x^7 + x^5 + 2x^6 + 2x^4 + 3x^4 + 3x^2 + x^3 + x  3x^2  3

Simplify by combining like terms:
 x^7 + 2x^6 + x^5 + 5x^4 + x^3 + x  3
Result
Therefore, the expanded form of (x^5 + 2x^4 + 3x^2 + x  3)(x^2 + 1) is x^7 + 2x^6 + x^5 + 5x^4 + x^3 + x  3.