Simplify x^2 + 3^2 + 1.

Step-by-step solution

1. 1

   Evaluate the numeric power before touching anything else. Order of operations puts exponents ahead of addition, and $3^{2}$ involves no variable at all, so it is simply a number:

   $3^{2}=3\times3=9$

   A frequent error is reading $3^{2}$ as $3\times2=6$; the exponent counts how many copies are multiplied, not a factor.

2. 2

   Rewrite the expression with that value in place.

   $x^{2}+3^{2}+1=x^{2}+9+1$

3. 3

   Combine only the constants. $9$ and $1$ are like terms (both pure numbers), so they add. The term $x^{2}$ is not like them — it carries a variable — so it cannot join in:

   $9+1=10\qquad\Longrightarrow\qquad x^{2}+10$

   In particular $x^{2}+9$ does not become $x^{2+9}$ or $9x^{2}$; addition never merges a variable term with a constant.

4. 4

   Check at two values. At $x=2$: original $4+9+1=14$, simplified $4+10=14$ ✓. At $x=-5$: $25+9+1=35$ and $25+10=35$ ✓. Agreement holds at every integer from $-20$ to $19$ ✓.

5. 5

   Note what the form tells you. $x^{2}+10$ is a parabola shifted up $10$ units, so its minimum value is $10$ at $x=0$ and it is never zero — $x^{2}+10=0$ would need $x^{2}=-10$, impossible for real $x$. That also means $x^{2}+3^{2}+1>0$ for every real $x$.