Let's go through the subtraction of the two given polynomials step-by-step. The polynomials we're dealing with are:
[tex]\[ P_1(x) = 5x^2 + 2x - 6 \][/tex]
[tex]\[ P_2(x) = 3x^2 - 6x + 2 \][/tex]
We need to subtract [tex]\( P_2(x) \)[/tex] from [tex]\( P_1(x) \)[/tex]:
[tex]\[ P(x) = P_1(x) - P_2(x) = (5x^2 + 2x - 6) - (3x^2 - 6x + 2) \][/tex]
To perform the subtraction, we subtract the corresponding coefficients of each term:
- For the [tex]\( x^2 \)[/tex] term:
[tex]\[ 5x^2 - 3x^2 = 2x^2 \][/tex]
- For the [tex]\( x \)[/tex] term:
[tex]\[ 2x - (-6x) = 2x + 6x = 8x \][/tex]
- For the constant term:
[tex]\[ -6 - 2 = -8 \][/tex]
Thus, after performing the subtraction, the resulting polynomial is:
[tex]\[ P(x) = 2x^2 + 8x - 8 \][/tex]
Since all the steps have led us to another polynomial, the result [tex]\( P(x) = 2x^2 + 8x - 8 \)[/tex] is indeed a polynomial and confirms that the subtraction of these two given polynomials results in another polynomial.
Therefore, among the given choices, the correct answer is:
[tex]\[ 2x^2 + 8x - 8 \text{, will be a polynomial} \][/tex]
