can you please solve for me the following problem:

                                                                               

what is the lowest polynomial in 'x'should be added to(x^{2_+x+1)(x2_{-x+1)to make it aperfect square}}

Dear Rahul i didn't exactly understand the meaning of 'lowest polynomial'. However, i consider it as a polynomial of lowest degree and solve the problem as below:

(x^{2_+x+1)(x2_{-x+1) =}} x⁴ + x²+ 1

Thus, if we add x² to above term it becomes a perfect square i.e.

(x⁴ + x²+ 1) + x² = x⁴ + 2x²+ 1 = (x²+ 1)²