Assertion : `3x^(2)+x-1=(x+1)(3x-2)+1` .
Reason : If `p(x)` and `g(x)` are two polynomials such that degree of `p(x)ge` degree of `g(x)` and `g(x)ge0` then we can find polynomials `q(x)` and `r(x)` such that `p(x)=g(x).q(x)+r(x)`, where `r(x)=0` or degree of `r(x)lt` degree of `g(x)` .

Options

A. If both assertion and reason are true and reason is the correct explanation of assertion .
B. If both assertion and reason are true but reason is not the correct explanation of assertion .
C. If assertion is true but reason is false .
D. If assertion is false but reason is true .

Answer

To solve the assertion and reason question, we need to verify the assertion and understand the reason provided.

### Step 1: Verify the Assertion
We need to check if the equation given in the assertion is true:
\[ 3x^2 + x - 1 = (x + 1)(3x - 2) + 1 \]

First, we will expand the right-hand side:
1. Expand \( (x + 1)(3x - 2) \):
\[ (x + 1)(3x - 2) = x \cdot 3x + x \cdot (-2) + 1 \cdot 3x + 1 \cdot (-2) \]
\[ = 3x^2 - 2x + 3x - 2 = 3x^2 + x - 2 \]

2. Now, add 1 to the result:
\[ 3x^2 + x - 2 + 1 = 3x^2 + x - 1 \]

Thus, we have:
\[ 3x^2 + x - 1 = 3x^2 + x - 1 \]
This confirms that the assertion is true.

### Step 2: Understand the Reason
The reason states that if \( p(x) \) and \( g(x) \) are two polynomials such that the degree of \( p(x) \) is greater than or equal to the degree of \( g(x) \), and \( g(x) \) is greater than or equal to 0, then we can find polynomials \( q(x) \) and \( r(x) \) such that:
\[ p(x) = g(x) \cdot q(x) + r(x) \]
where \( r(x) = 0 \) or the degree of \( r(x) \) is less than the degree of \( g(x) \).

This is a statement of the polynomial division algorithm, which is indeed true.

### Conclusion
Since both the assertion and the reason are true, and the reason correctly explains the assertion, we conclude that both the assertion and the reason are true.

### Final Answer
Both the assertion and reason are true, and the reason is the correct explanation of the assertion.

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