Sure, let's find the sum of the given polynomials step-by-step. The polynomials given are:
[tex]\[ P(x) = x^2 - 3x \][/tex]
[tex]\[ Q(x) = -2x^2 + 5x - 3 \][/tex]
To find the sum, we need to combine like terms. Like terms are the terms that contain the same power of [tex]\(x\)[/tex]. Let's align the terms according to their powers:
[tex]\[ \left(x^2 - 3x + 0\right) + \left(-2x^2 + 5x - 3\right) \][/tex]
Now, add the coefficients of like terms:
1. [tex]\(x^2\)[/tex] terms:
[tex]\[ 1x^2 + (-2x^2) = 1 - 2 = -1 \][/tex]
So, the coefficient of [tex]\(x^2\)[/tex] is [tex]\(-1\)[/tex].
2. [tex]\(x\)[/tex] terms:
[tex]\[ -3x + 5x = -3 + 5 = 2 \][/tex]
So, the coefficient of [tex]\(x\)[/tex] is [tex]\(2\)[/tex].
3. Constant term:
[tex]\[ 0 + (-3) = -3 \][/tex]
So, the constant term is [tex]\(-3\)[/tex].
Combining these results, the sum of the polynomials is:
[tex]\[ P(x) + Q(x) = -1x^2 + 2x - 3 \][/tex]
So, the sum of the given polynomials in standard form is:
[tex]\[ -x^2 + 2x - 3 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{-x^2 + 2x - 3} \][/tex]
