Example 5 Solve the equation
$$
\sqrt{x^{2}+3 x+7}-\sqrt{x^{2}-x+10}=1 \text {. }
$$
Solving irrational equations by the usual method (transposition, squaring, simplification) can be quite cumbersome. Observing the characteristics of the given equation, one can think of the difference of squares formula. Construct the identity
$$
\left(x^{2}+3 x+7\right)-\left(x^{2}-x+10\right)=4 x-3 \text {. }
$$

Dividing both sides of the above equation by both sides of the original equation, we get
$$
\sqrt{x^{2}+3 x+7}+\sqrt{x^{2}-x+10}=4 x-3 \text {. }
$$

Adding this equation to the original equation, we get
$$
\sqrt{x^{2}+3 x+7}=2 x-1 \text {. }
$$

Solving this, we get $x=3$.