To solve this problem, we first need to understand the sets $P$ and $Q$, and then determine the number of elements in the intersection $P \cap Q$.

Set $P$

The problem states that $P = \{x = [-1,1]: +21=0\}$. It appears there is a typographical error or confusion in the notation. However, interpreting it correctly, it seems to be referring to the interval $x \in [-1, 1]$. So:

$P = \{ x \in \mathbb{R} : -1 \leq x \leq 1 \}$

Set $Q$

The set $Q = \{ x \in \mathbb{Z} : x^2 - 3x - 10 \leq 0 \}$.

To find $Q$, we solve the inequality $x^2 - 3x - 10 \leq 0$:

1. Factor the quadratic equation: $x^2 - 3x - 10 = (x - 5)(x + 2) = 0$

2. The roots are $x = 5$ and $x = -2$. The parabola opens upwards, so the quadratic inequality $x^2 - 3x - 10 \leq 0$ holds between the roots:

   $-2 \leq x \leq 5$

3. Since $Q$ consists of integers: $Q = \{ x \in \mathbb{Z} : -2 \leq x \leq 5 \} = \{-2, -1, 0, 1, 2, 3, 4, 5\}$

Intersection $P \cap Q$

Now, the intersection $P \cap Q$ is the set of elements that are in both $P$ and $Q$.

Since $P = [-1, 1]$ and $Q = \{-2, -1, 0, 1, 2, 3, 4, 5\}$, the intersection $P \cap Q$ includes only those elements of $Q$ that are within the interval $[-1, 1]$:

$P \cap Q = \{-1, 0, 1\}$

The number of elements in $P \cap Q$ is $3$.

Final Answer

The correct answer is:

(4) 3

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Would you like any more details or have any further questions?

Here are some related questions:

1. How do you solve a quadratic inequality in general?
2. What is the intersection of two sets, and how is it calculated?
3. What are the properties of quadratic functions and their graphs?
4. How can you factorize a quadratic expression?
5. What is the significance of integer sets in mathematics?

Tip: When solving quadratic inequalities, always remember to first find the roots by solving the equality and then analyze the sign of the quadratic expression between the roots to determine where the inequality holds.