we have
\(4x^{2} +11x+6\)
Equate the expression to zero
\(4x^{2} +11x+6=0\)
Group terms that contain the same variable, and move the constant to the opposite side of the equation
\(4x^{2} +11x=-6\)
Factor the leading coefficient
\(4(x^{2} +(11x/4))=-6\)
Complete the square. Remember to balance the equation by adding the same constants to each side
\(4(x^{2} +(11x/4)+(121/64))=-6+(121/16)\)
\(4(x^{2} +(11x/4)+(121/64))=(25/16)\)
\((x^{2} +(11x/4)+(121/64))=(25/64)\)
Rewrite as perfect squares
\((x+(11/8))^{2}=(25/64)\)
Square root both sides
\(x+(11/8)=(+/-)\sqrt{\frac{25}{64}}\)
\(x=(-11/8)(+/-)\frac{5}{8}\)
\(x=(-11/8)+\frac{5}{8}=-\frac{6}{8}=-\frac{3}{4}\)
\(x=(-11/8)-\frac{5}{8}=-\frac{16}{8}=-2\)
therefore
\(4x^{2} +11x+6=4(x+\frac{3}{4})(x+2)=(4x+3)((x+2)\)
the answer is
\((4x+3)((x+2)\)
