we have
\(3x^{2} -8x+5\)
Equate the expression to zero to find the roots
\(3x^{2} -8x+5=0\)
Group terms that contain the same variable, and move the constant to the opposite side of the equation
\(3x^{2} -8x=-5\)
Factor the leading coefficient
\(3(x^{2} -(8x/3))=-5\)
Complete the square. Remember to balance the equation by adding the same constants to each side.
\(3(x^{2} -(8x/3)+(16/9))=-5+(16/3)\)
\(3(x^{2} -(8x/3)+(16/9))=(1/3)\)
Rewrite as perfect squares
\(3(x-(4x/3))^{2}=(1/3)\)
\((x-(4x/3))^{2}=(1/9)\)
square roots both sides
\((x-\frac{4}{3})=(+/-) \sqrt{\frac{1}{9}}\)
\(x=\frac{4}{3}(+/-) \frac{1}{3}\)
the roots are
\(x=\frac{4}{3}+ \frac{1}{3}=\frac{5}{3}\)
\(x=\frac{4}{3}- \frac{1}{3}=\frac{3}{3}=1\)
so
\(3x^{2} -8x+5=3(x-\frac{5}{3})(x-1)=(3x-5)(x-1)\)
therefore
the answer is the option
\((3x-5)(x-1)\)
