Solve (a-2)^2+4(a-2)<0

$$( a - 2 ) ^ { 2 } + 4 ( a - 2 ) < 0$$

Solve for a

$$a\in \left(-2,2\right)$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(a-2\right)^{2}.

$$a^{2}-4a+4+4\left(a-2\right)<0$$

Use the distributive property to multiply 4 by a-2.

$$a^{2}-4a+4+4a-8<0$$

Combine -4a and 4a to get 0.

$$a^{2}+4-8<0$$

Subtract 8 from 4 to get -4.

$$a^{2}-4<0$$

Add 4 to both sides.

$$a^{2}<4$$

Calculate the square root of 4 and get 2. Rewrite 4 as 2^{2}.

$$a^{2}<2^{2}$$

Inequality holds for |a|<2.

$$|a|<2$$

Rewrite |a|<2 as a\in \left(-2,2\right).

$$a\in \left(-2,2\right)$$