Question
$$\frac{1+x}{2}+\frac{3-x}{4} = -x^2-4x-1$$
Answer
$$ \begin{matrix}x_1 = -\dfrac{ 17 }{ 8 }-\dfrac{\sqrt{ 145 }}{ 8 } & x_2 = -\dfrac{ 17 }{ 8 }+\dfrac{\sqrt{ 145 }}{ 8 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{1+x}{2}+\frac{3-x}{4} &= -x^2-4x-1&& \text{multiply ALL terms by } \color{blue}{ 4 }. \\[1 em]4 \cdot \frac{1+x}{2}+4\frac{3-x}{4} &= -4x^2-4\cdot4x-4\cdot1&& \text{cancel out the denominators} \\[1 em]2x+2+3-x &= -4x^2-16x-4&& \text{simplify left side} \\[1 em]x+5 &= -4x^2-16x-4&& \text{move all terms to the left hand side } \\[1 em]x+5+4x^2+16x+4 &= 0&& \text{simplify left side} \\[1 em]4x^2+17x+9 &= 0&& \\[1 em] \end{aligned} $$
$ 4x^{2}+17x+9 = 0 $ is a quadratic equation.
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