Find the Determinant [[2s+3,-3],[4/(s+1)+s+2,s^2]]

Problem

det(2*s+3)

Solution

1. Apply the determinant formula for a 2×2 matrix, which is det(a)=a*d−b*c

a=2*s+3

b=−3

c=4/(s+1)+s+2

d=s2()

2. Set up the product of the diagonal elements and the off-diagonal elements.

det(=)*(2*s+3)*(s2)−(−3)*(4/(s+1)+s+2)

3. Distribute the terms in the first part of the expression.

(2*s+3)*(s2)=2*s3+3*s2()

4. Distribute the constant −(−3)=3 into the second part of the expression.

3*(4/(s+1)+s+2)=12/(s+1)+3*s+6

5. Combine the results to form the full polynomial expression.

det(=)*2*s3+3*s2+3*s+6+12/(s+1)

6. Find a common denominator to write the expression as a single fraction if necessary.

det(=)((2*s3+3*s2+3*s+6)*(s+1)+12)/(s+1)

7. Expand the numerator by multiplying the polynomial by (s+1)

(2*s3+3*s2+3*s+6)*(s+1)=2*s4+2*s3+3*s3+3*s2+3*s2+3*s+6*s+6

2*s4+5*s3+6*s2+9*s+6

8. Add the remaining constant to the expanded numerator.

2*s4+5*s3+6*s2+9*s+6+12=2*s4+5*s3+6*s2+9*s+18

Final Answer

det(2*s+3)=(2*s4+5*s3+6*s2+9*s+18)/(s+1)

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