Step 1 :a) \( x^4 + 4 = (x^2 + 2x + 2)(x^2 - 2x + 2) \) in \( \mathbb{Z}_{5}[x] \)
Step 2 :b) \( x^3+ 2x^2 + 2x + 1 = (x+1)(x^2+x+1) \) in \( \mathbb{Z}_{7}[x] \)
Step 3 :c) \( x^2+x+1 = (x^2+x+1) \) in \( \mathbb{Z}_{2}[x] \)
Step 4 :2) \( x^3+2x+3 = (x^3+2x+3) \), irreducible in \( \mathbb{Z}_{5}[x] \)
Step 5 :3) \( 2x^3+x^2+2x+2 = (x^3+(1/2)x^2+x+1) \), irreducible in \( \mathbb{Z}_{5}[x] \)