Simplify (a - 2b)^2 + 8ab.

Step-by-step solution

1. 1

   Expand the square. Use $(p-q)^2=p^2-2pq+q^2$ with $p=a$, $q=2b$:

   $(a-2b)^2=a^2-2(a)(2b)+(2b)^2=a^2-4ab+4b^2$

2. 2

   Add the loose term and combine like terms. Only the $ab$ terms interact:

   $a^2-4ab+4b^2+8ab=a^2+4ab+4b^2$

   since $-4ab+8ab=4ab$.

3. 3

   Recognise the result as a perfect square. Now $a^2$, $4b^2=(2b)^2$ and the middle term $2\cdot a\cdot 2b=4ab$ all match the square-of-a-sum pattern:

   $a^2+4ab+4b^2=(a+2b)^2$

4. 4

   Note the structural reason. The two squares $(a\pm 2b)^2$ differ by exactly $2\cdot 2\cdot(a)(2b)=8ab$:

   $(a+2b)^2-(a-2b)^2=8ab$

   so adding $8ab$ to the difference-square is guaranteed to produce the sum-square. This identity is the general fact behind the problem.

5. 5

   Check numerically. At $a=3$, $b=1$: the original is $(3-2)^2+24=1+24=25$, and $(3+2)^2=25$ $\checkmark$.