If the quadratic polynomials defined on real coefficient
`P(x)=a_(1)x^(2)+2b_(1)x+c_(1)` and `Q(x)=a_(2)x^(2)+2b_(2)x+c_(2)` take positive values `AA x in R`, what can we say for the trinomial `g(x)=a_(1)a_(2)x^(2)+b_(1)b_(2)x+c_(1)c_(2)` ?

Options

A. `g(x)` takes positive values only.
B. `g(x)` takes negative values only.
C. `g(x)` can takes positive as well as negative values.
D. Nothing definite can be said about `g(x)`.

Explanation

To solve the problem, we need to analyze the given quadratic polynomials \( P(x) \) and \( Q(x) \) and their properties, and then determine the behavior of the trinomial \( g(x) \).

### Step 1: Analyze the Quadratic Polynomials
The polynomials are given as:
\[ P(x) = a_1 x^2 + 2b_1 x + c_1 \]
\[ Q(x) = a_2 x^2 + 2b_2 x + c_2 \]
Since both \( P(x) \) and \( Q(x) \) take positive values for all \( x \in \mathbb{R} \), we can conclude that:
1. The leading coefficients \( a_1 \) and \( a_2 \) must be positive (i.e., \( a_1 > 0 \) and \( a_2 > 0 \)).
2. The discriminants of both polynomials must be negative.

### Step 2: Calculate the Discriminants
The discriminant \( D \) of a quadratic polynomial \( ax^2 + bx + c \) is given by \( D = b^2 - 4ac \).

For \( P(x) \):
\[ D_1 = (2b_1)^2 - 4a_1c_1 = 4b_1^2 - 4a_1c_1 < 0 \implies b_1^2 < a_1c_1 \quad \text{(Equation 1)} \]

For \( Q(x) \):
\[ D_2 = (2b_2)^2 - 4a_2c_2 = 4b_2^2 - 4a_2c_2 < 0 \implies b_2^2 < a_2c_2 \quad \text{(Equation 2)} \]

### Step 3: Multiply the Inequalities
From Equation 1 and Equation 2, we multiply both sides:
\[ (b_1^2)(b_2^2) < (a_1c_1)(a_2c_2) \]
This gives us:
\[ a_1a_2c_1c_2 > b_1^2b_2^2 \]

### Step 4: Analyze the Trinomial \( g(x) \)
The trinomial is defined as:
\[ g(x) = a_1a_2x^2 + b_1b_2x + c_1c_2 \]
We need to find the discriminant of \( g(x) \):
\[ D_g = (b_1b_2)^2 - 4(a_1a_2)(c_1c_2) \]
Substituting from our previous result:
\[ D_g = b_1^2b_2^2 - 4a_1a_2c_1c_2 \]
Using the inequality we derived:
\[ D_g < b_1^2b_2^2 - 4b_1^2b_2^2 = -3b_1^2b_2^2 \]
Since \( b_1^2b_2^2 > 0 \), it follows that:
\[ D_g < 0 \]

### Step 5: Conclusion
Since the discriminant \( D_g \) is negative, the trinomial \( g(x) \) has no real roots, and because the leading coefficient \( a_1a_2 > 0 \), it implies that \( g(x) \) takes positive values for all \( x \in \mathbb{R} \).

Thus, the answer is:
**Option A: \( g(x) \) takes positive values only.**