Introduction:
To factorize the expression 2x^2 + 7x + 3 using the factor theorem, we need to understand the concept of the factor theorem and apply it step by step. The factor theorem states that if a polynomial expression P(x) is divided by (x - a) and the remainder is zero, then (x - a) is a factor of P(x).

Step 1: Finding the possible factors:
In this case, we need to find the factors of the constant term, which is 3. The factors of 3 are ±1 and ±3. These are the possible values for 'a' in the factor theorem.

Step 2: Applying the factor theorem:
We will substitute each possible value of 'a' into the polynomial expression and check if the remainder is zero. If the remainder is zero, it means that (x - a) is a factor of the expression.

For (x - 1):
P(1) = 2(1)^2 + 7(1) + 3 = 2 + 7 + 3 = 12 (not zero)

For (x + 1):
P(-1) = 2(-1)^2 + 7(-1) + 3 = 2 - 7 + 3 = -2 (not zero)

For (x - 3):
P(3) = 2(3)^2 + 7(3) + 3 = 18 + 21 + 3 = 42 (not zero)

For (x + 3):
P(-3) = 2(-3)^2 + 7(-3) + 3 = 18 - 21 + 3 = 0 (zero)

Step 3: Identifying the factor:
Since (x + 3) gives a remainder of zero, we can conclude that (x + 3) is a factor of the polynomial expression 2x^2 + 7x + 3.

Step 4: Dividing the polynomial:
Using synthetic division or long division, we divide 2x^2 + 7x + 3 by (x + 3) to find the other factor.

The division gives us:

2x + 1

___________________
x + 3 | 2x^2 + 7x + 3
- (2x^2 + 6x)
____________
x + 3
- (x + 3)
___________
0

Step 5: Writing the factored form:
We have factored 2x^2 + 7x + 3 as (x + 3)(2x + 1).

Conclusion:
Using the factor theorem, we were able to factorize the expression 2x^2 + 7x + 3 as (x + 3)(2x + 1). The factor theorem helps us identify the factors by checking if the remainder is zero when substituting possible values of 'a'. This method allows us to simplify complex polynomial expressions and understand their underlying factors.

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