Evaluate: (a-2)(2a+1)

Factor: $\left(a-2\right)\left(2a+1\right)$

Multiply and combine like terms.

$$2a^{2}-3a-2$$

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

$$p+q=-3$$ $$pq=2\left(-2\right)=-4$$

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 $-4$.

$$1,-4$$ $$2,-2$$

Calculate the sum for each pair.

$$1-4=-3$$ $$2-2=0$$

The solution is the pair that gives sum $-3$.

$$p=-4$$ $$q=1$$

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

$$\left(2a^{2}-4a\right)+\left(a-2\right)$$

Factor out $2a$ in $2a^{2}-4a$.

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

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

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