Simplify (x + 4)^2 - (x^2 + 8x + 16).

Step-by-step solution

1. 1

   Expand the square using the perfect-square formula. The identity $(a+b)^{2}=a^{2}+2ab+b^{2}$ with $a=x$, $b=4$ gives

   $(x+4)^{2}=x^{2}+2(x)(4)+4^{2}=x^{2}+8x+16.$

   The middle term is $2\cdot x\cdot 4=8x$, not $4x$ — forgetting the factor of $2$ is by far the most common expansion error here.

2. 2

   Notice that the bracket is the same trinomial. The expression now reads

   $\left(x^{2}+8x+16\right)-\left(x^{2}+8x+16\right),$

   which is a quantity minus itself. Recognising this immediately is faster than grinding through the algebra, and it explains the answer before you compute it: the problem is really asking whether $x^{2}+8x+16$ is the expansion of $(x+4)^{2}$.

3. 3

   Distribute the minus sign over every term. Writing it out to be safe:

   $x^{2}+8x+16-x^{2}-8x-16.$

   All three signs inside the bracket flip. Changing only the first term — a very common slip — would leave $16x+32$ instead of $0$.

4. 4

   Combine like terms. Grouping by degree:

   $(x^{2}-x^{2})+(8x-8x)+(16-16)=0+0+0=0.$

   The expression simplifies to the constant $0$.

5. 5

   Interpret the result. The value is $0$ for every real $x$, so this is an identity, not an equation to solve. Checking at $x=3$: $(3+4)^{2}=49$ and $9+24+16=49$, difference $0$ ✓; at $x=-1$: $9$ and $1-8+16=9$, difference $0$ ✓. Two test values agreeing is expected — the two expressions are literally the same polynomial.