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We are given the following information from the image:

The quadratic polynomial \( p(x) \) satisfies:
1. \( p(x) \) can be positive or zero for all real numbers.
2. \( p(1) = 0 \) and \( p(2) = 2 \).

We need to find the quadratic polynomial \( p(x) \).

### Step 1: General form of a quadratic polynomial
The general form of a quadratic polynomial is:
\[
p(x) = ax^2 + bx + c
\]
where \( a \), \( b \), and \( c \) are constants.

### Step 2: Use the conditions \( p(1) = 0 \) and \( p(2) = 2 \)

Substitute \( x = 1 \) and \( p(1) = 0 \):
\[
a(1)^2 + b(1) + c = 0 \quad \Rightarrow \quad a + b + c = 0 \quad \text{(Equation 1)}
\]

Substitute \( x = 2 \) and \( p(2) = 2 \):
\[
a(2)^2 + b(2) + c = 2 \quad \Rightarrow \quad 4a + 2b + c = 2 \quad \text{(Equation 2)}
\]

### Step 3: Analyze the positivity condition
Since the quadratic is positive or zero for all real numbers, its graph must not cross below the x-axis. This suggests the quadratic has a vertex on or above the x-axis, and the coefficient \( a \) must be positive (so it opens upwards). Additionally, \( p(1) = 0 \) implies that \( x = 1 \) is a root.

Thus, we assume the quadratic can be written in factored form as:
\[
p(x) = a(x - 1)^2
\]
Expanding this:
\[
p(x) = a(x^2 - 2x + 1) = ax^2 - 2ax + a
\]

### Step 4: Use the condition \( p(2) = 2 \)

Substitute \( x = 2 \) into the factored form:
\[
p(2) = a(2 - 1)^2 = a(1)^2 = a
\]
Since \( p(2) = 2 \), we get:
\[
a = 2
\]

### Step 5: Final quadratic equation

Substitute \( a = 2 \) into the equation \( p(x) = a(x - 1)^2 \):
\[
p(x) = 2(x - 1)^2
\]
Expanding it gives:
\[
p(x) = 2(x^2 - 2x + 1) = 2x^2 - 4x + 2
\]

Thus, the quadratic polynomial is:
\[
p(x) = 2x^2 - 4x + 2
\]

### Final answer:
The quadratic polynomial is \( p(x) = 2x^2 - 4x + 2 \).

Would you like more details or have any other questions? 

Here are some related questions:
1. How do you verify if a quadratic polynomial is always positive?
2. What is the significance of the vertex in determining the shape of a parabola?
3. How can you find the vertex of a quadratic function from its standard form?
4. How does the discriminant help determine the number of roots of a quadratic equation?
5. What are the applications of quadratic polynomials in real-world problems?

**Tip:** Always remember that if a quadratic polynomial is always positive or zero, its vertex must lie at or above the x-axis, and its leading coefficient must be positive.

The quadratic polynomial p(x) has the following properties: p(x) can be positive or zero for all real numbers. p(1) = 0 and p(2) = 2. Then find the quadratic polynomial.

Learn how to find the quadratic polynomial p(x) given conditions p(1)=0 and p(2)=2, using the general form of a quadratic equation and positivity conditions for all real numbers.

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