Problem 4 A polynomial f ( *) has the factor - square property ( or ESP ) if f ( * ) is a factor of f ( 20 2 ) . For instance ,* 9 ( 20 ) = * - 1 and h ( a ) = ac have ESP , but K ( 20 ) = * + 2 does not . Reason : * - 1 is a factor of 2 2 - 1 , and * is a factor of *2 , but a + 2 is not a factor of 2 2 + 2 . Multiplying by a nonzero constant " preserves " ISP , so we restrict attention to poly - nomials that are monic ( i.e ., have I as highest- degree coefficient ) . What is the pattern to these ESP polynomials ? To make progress on this general problem , investigate the following questions and justify your answers . ( a ) Are * and * - I the only monic polynomials of degree I with ESP ? ( b ) Check that 2 2 , 2 2 - 1 , 2 2 - * , and a 2 + 2 + 1 all have ESP. Determine all the monic degree 2 polynomials with ESP . ( c ) Some of our examples are products of ESP polynomials of smaller degree . For instance , * _ and a_ _ * come from degree I cases . However , 2 2 - 1 and 2 2 + 2 + 1 are new , not expressible as a product of two smaller ISP polynomials . Are there monic ESP polynomials of degree & that are new ( not built from ESP polynomials of smaller degree ) ?' Are there such examples of degree 4 ? ( d ) The examples written above all had integer coefficients . Do answers change if we allow polynomials whose coefficients are allowed to be any real numbers ? Or if we allow polynomials whose coefficients are complex numbers ?'