$$a+b=2$$ $$ab=8\left(-3\right)=-24$$
Since $ab$ is negative, $a$ and $b$ have the opposite signs. Since $a+b$ is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product $-24$.$$-1,24$$ $$-2,12$$ $$-3,8$$ $$-4,6$$
Calculate the sum for each pair.$$-1+24=23$$ $$-2+12=10$$ $$-3+8=5$$ $$-4+6=2$$
The solution is the pair that gives sum $2$.$$a=-4$$ $$b=6$$
Rewrite $8x^{2}+2x-3$ as $\left(8x^{2}-4x\right)+\left(6x-3\right)$.$$\left(8x^{2}-4x\right)+\left(6x-3\right)$$
Factor out $4x$ in the first and $3$ in the second group.$$4x\left(2x-1\right)+3\left(2x-1\right)$$
Factor out common term $2x-1$ by using distributive property.$$\left(2x-1\right)\left(4x+3\right)$$
To find equation solutions, solve $2x-1=0$ and $4x+3=0$.$$x=\frac{1}{2}$$ $$x=-\frac{3}{4}$$