Expand and simplify (√2 + √3 + √5)².

Step-by-step solution

1. 1

   Use the three-term square formula. For any $a,b,c$,

   $(a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2ab+2bc+2ca$

   There are three cross terms, not one — forgetting a pair is the standard mistake with three-term squares.

2. 2

   Square each term. Squaring a square root removes it entirely:

   $\left(\sqrt2\right)^{2}+\left(\sqrt3\right)^{2}+\left(\sqrt5\right)^{2}=2+3+5=10$

3. 3

   Form the three cross terms. Each is twice the product of a pair, and $\sqrt a\cdot\sqrt b=\sqrt{ab}$:

   $2\sqrt2\sqrt3=2\sqrt6,\qquad 2\sqrt3\sqrt5=2\sqrt{15},\qquad 2\sqrt5\sqrt2=2\sqrt{10}$

4. 4

   Add everything. None of $\sqrt6$, $\sqrt{10}$, $\sqrt{15}$ can be simplified further or combined with each other, since $6$, $10$ and $15$ are square-free and distinct:

   $\boxed{10+2\sqrt6+2\sqrt{10}+2\sqrt{15}}$

5. 5

   Check numerically. $\sqrt2+\sqrt3+\sqrt5=1.41421+1.73205+2.23607=5.38233$, and $5.38233^{2}=28.9695$. The answer gives $10+2(2.44949)+2(3.16228)+2(3.87298)=10+4.89898+6.32456+7.74597=28.9695$ ✓.