Solve 12a^2+3a-8a-2

$$12 a ^ { 2 } + 3 a - 8 a - 2$$

Factor

$$\left(3a-2\right)\left(4a+1\right)$$

Solution Steps

Multiply and combine like terms.

$$12a^{2}-5a-2$$

Factor the expression by grouping. First, the expression needs to be rewritten as 12a^{2}+pa+qa-2. To find p and q, set up a system to be solved.

$$p+q=-5 pq=12\left(-2\right)=-24$$

Since pq is negative, p and q have the opposite signs. Since p+q is negative, the negative number has greater absolute value than the positive. 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 -5.

$$p=-8 q=3$$

Rewrite 12a^{2}-5a-2 as \left(12a^{2}-8a\right)+\left(3a-2\right).

$$\left(12a^{2}-8a\right)+\left(3a-2\right)$$

Factor out 4a in 12a^{2}-8a.

$$4a\left(3a-2\right)+3a-2$$

Factor out common term 3a-2 by using distributive property.

$$\left(3a-2\right)\left(4a+1\right)$$

Evaluate

$$\left(3a-2\right)\left(4a+1\right)$$

Solution Steps

Combine 3a and -8a to get -5a.

$$12a^{2}-5a-2$$