Solve for: x ^ 2+4 x+3 = 0

Expression: $$x ^ { 2 } + 4 x + 3 = 0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as $x^{2}+ax+bx+3$. To find $a$ and $b$, set up a system to be solved.

$$a+b=4$$ $$ab=1\times 3=3$$

Since $ab$ is positive, $a$ and $b$ have the same sign. Since $a+b$ is positive, $a$ and $b$ are both positive. The only such pair is the system solution.

$$a=1$$ $$b=3$$

Rewrite $x^{2}+4x+3$ as $\left(x^{2}+x\right)+\left(3x+3\right)$.

$$\left(x^{2}+x\right)+\left(3x+3\right)$$

Factor out $x$ in the first and $3$ in the second group.

$$x\left(x+1\right)+3\left(x+1\right)$$

Factor out common term $x+1$ by using distributive property.

$$\left(x+1\right)\left(x+3\right)$$

To find equation solutions, solve $x+1=0$ and $x+3=0$.

$$x=-1$$ $$x=-3$$