If the domain of the function $f(x) = \frac{1}{\sqrt{10 + 3x - x^2}} + \frac{1}{\sqrt{x + |x|}}$ is $(a,\ b)$, then $(1 + a)^2 + b^2$ is equal to:

Step 1: Determine the domain for the first term

For the first term, $\frac{1}{\sqrt{10 + 3x - x^2}}$, the expression under the square root must be strictly positive. This means $10 + 3x - x^2 > 0$. We can rewrite this inequality as $x^2 - 3x - 10 < 0$. Factoring the quadratic expression, we get $(x - 5)(x + 2) < 0$. This inequality holds true when $x$ is between the roots, so $-2 < x < 5$.

Step 2: Determine the domain for the second term

For the second term, $\frac{1}{\sqrt{x + |x|}}$, the expression under the square root must also be strictly positive. This means $x + |x| > 0$. We consider two cases for $|x|$: If $x \ge 0$, then $|x| = x$, so $x + x > 0 \implies 2x > 0 \implies x > 0$. If $x < 0$, then $|x| = -x$, so $x + (-x) > 0 \implies 0 > 0$, which is false. Therefore, the condition $x + |x| > 0$ is satisfied only when $x > 0$.

Step 3: Find the intersection of the domains

The domain of the entire function $f(x)$ is the intersection of the domains of its individual terms. We found that for the first term, the domain is $(-2, 5)$, and for the second term, the domain is $(0, \infty)$. The intersection of these two intervals is $(0, 5)$.

Step 4: Identify 'a' and 'b' and calculate the final expression

From the combined domain $(0, 5)$, we can identify $a = 0$ and $b = 5$. Now, we substitute these values into the expression $(1 + a)^2 + b^2$. This gives us $(1 + 0)^2 + 5^2 = 1^2 + 5^2 = 1 + 25 = 26$.

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