Let's simplify the given polynomial and then evaluate it for [tex]\( x = 3 \)[/tex].
### Step 1: Simplify the Polynomial
The given polynomial is:
[tex]\[ x^2 + 2x - 3 - 2x^2 + x + 4 \][/tex]
First, let's combine the like terms. The polynomial has the terms:
[tex]\[ x^2 \][/tex]
[tex]\[ 2x - 2x^2 + x \][/tex]
[tex]\[ -3 + 4 \][/tex]
Combining these, we get:
[tex]\[ (x^2 - 2x^2) + (2x + x) + (-3 + 4) \][/tex]
Let's simplify each type separately:
[tex]\[ (x^2 - 2x^2) = -x^2 \][/tex]
[tex]\[ (2x + x) = 3x \][/tex]
[tex]\[ (-3 + 4) = 1 \][/tex]
So, the simplified polynomial is:
[tex]\[ -x^2 + 3x + 1 \][/tex]
### Step 2: Evaluate the Simplified Polynomial for [tex]\( x = 3 \)[/tex]
Now, we need to evaluate this simplified polynomial at [tex]\( x = 3 \)[/tex]:
[tex]\[ -x^2 + 3x + 1 \][/tex]
Substitute [tex]\( x = 3 \)[/tex] into the polynomial:
[tex]\[ -(3)^2 + 3(3) + 1 \][/tex]
Calculate each term:
[tex]\[ -(3)^2 = -9 \][/tex]
[tex]\[ 3(3) = 9 \][/tex]
[tex]\[ 1 = 1 \][/tex]
Now add these values together:
[tex]\[ -9 + 9 + 1 = 1 \][/tex]
### Conclusion
The simplified polynomial is:
[tex]\[ -x^2 + 3x + 1 \][/tex]
And the value of the polynomial when [tex]\( x = 3 \)[/tex] is:
[tex]\[ 1 \][/tex]
So, the correct answer is:
[tex]\[ -x^2 + 3x + 1; 1 \][/tex]
