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