Question 609539
I assume that the equation is
arctan({{{3 + 4x + x^2}}}) = {{{-pi/4}}}<br>
arctan(something) represents "an angle whose tan ratio is 'something'". So 
arctan({{{3 + 4x + x^2}}})
represents an angle whose tan ratio is {{{3 + 4x + x^2}}}).
And the equation
arctan({{{3 + 4x + x^2}}}) = {{{-pi/4}}}
tells us that this angle is {{{-pi/4}}}
We should know that {{{tan(-pi/4) = -1}}}. Since {{{tan(-pi/4) = -1}}} and since the equation arctan({{{3 + 4x + x^2}}}) = {{{-pi/4}}} tells us that the tan of the same angle is also {{{3 + 4x+x^2}}} then {{{3 + 4x-x^2}}} and -1 must be equal to each other. Now we just have to solve:
{{{3 + 4x+x^2 = -1}}}
Since this is a quadratic equation we want one side to be zero. Adding 1 to each side we get:
{{{4 + 4x+x^2 = 0}}}
Next we factor (or use the Quadratic Formula). To make the factoring easier, I'm going to rearrange the terms:
{{{x^2 + 4x + 4 = 0}}}
This factors easily into:
(x+2)(x+2) = 0
From the Zero Product Property we know that one or more of the factors must be zero. So
x + 2 = 0
Solving we get
x = -2