Because the #x^4# coefficient is #1# we know the coefficient for the #x^2# terms in the factor will also be #1#:
#(x^2 )(x^2 )#
Because the constant is a negative and the coefficient for the #x# term is a negative we know the sign for the constants in the factors will have one positive and one negative:
#(x^2 + )(x^2 - )#
Now we need to determine the factors which multiply to -4 and also add to -3:
#1 xx -4 = -4#; #1 - 4 = -3# <- this IS the factor
#(x^2 + 1)(x^2 - 4)#
The factor #(x^2 - 4)# is a special form of the quadratic:
#color(red)(x)^2 - color(blue)(y)^2 = (color(red)(x) + color(blue)(y))(color(red)(x) - color(blue)(y))#
We can factor this term as:
#(x^2 + 1)(color(red)(x) + color(blue)(2))(color(red)(x) - color(blue)(2))#
