Phân tích mỗi đa thức sau thành nhân tử:

a) \(3{x^2} - \sqrt 3 x + \frac{1}{4}\)

\( = {\sqrt 3 ^2}.{x^2} - 2.\sqrt 3 x.\frac{1}{2} + {\left( {\frac{1}{2}} \right)^2}\)

\( = {\left( {\sqrt 3 x} \right)^2} - 2.\sqrt 3 x.\frac{1}{2} + {\left( {\frac{1}{2}} \right)^2}\)

\( = {\left( {\sqrt 3 x - \frac{1}{2}} \right)^2}\).

b) \({x^2} - x - {y^2} + y\)

\( = \left( {{x^2} - {y^2}} \right) - \left( {x - y} \right)\)

\( = \left( {x - y} \right)\left( {x + y} \right) - \left( {x - y} \right)\)

\( = \left( {x - y} \right)\left( {x + y - 1} \right)\).

c) \[{x^4} + {x^3} + 2{x^2} + x + 1\]

\[ = \left( {{x^4} + 2{x^2} + 1} \right) + \left( {{x^3} + x} \right)\]

\[ = \left[ {{{\left( {{x^2}} \right)}^2} + 2{x^2} + 1} \right] + \left( {{x^3} + x} \right)\]

\[ = {\left( {{x^2} + 1} \right)^2} + x\left( {{x^2} + 1} \right)\]

\[ = \left( {{x^2} + 1} \right)\left( {{x^2} + x + 1} \right)\].

d) \({x^3} + 2{x^2} + x - 16x{y^2}\)

\( = x\left( {{x^2} + 2x + 1 - 16{y^2}} \right)\)

\( = x\left[ {\left( {{x^2} + 2x + 1} \right) - {4^2}.{y^2}} \right]\)

\( = x\left[ {{{\left( {x + 1} \right)}^2} - {{\left( {4y} \right)}^2}} \right]\)

\( = x\left( {x - 4y + 1} \right)\left( {x + 4y + 1} \right)\).