To factorize the quadratic expression [tex]\(3x^2 - 8x + 5\)[/tex], we need to find two binomials whose product gives us the original expression.
Given a quadratic expression of the form [tex]\(ax^2 + bx + c\)[/tex]:
[tex]\[ 3x^2 - 8x + 5 \][/tex]
1. First, identify the coefficients: [tex]\(a = 3\)[/tex], [tex]\(b = -8\)[/tex], and [tex]\(c = 5\)[/tex].
2. We need to find two numbers that multiply to [tex]\(a \cdot c = 3 \cdot 5 = 15\)[/tex] and add up to [tex]\(b = -8\)[/tex].
3. The pairs of numbers that multiply to 15 are:
[tex]\[ (1, 15), (-1, -15), (3, 5), (-3, -5) \][/tex]
4. Among these pairs, the pair that adds up to -8 is [tex]\((-3, -5)\)[/tex].
5. Now rewrite the middle term [tex]\(-8x\)[/tex] using these numbers:
[tex]\[ 3x^2 - 3x - 5x + 5 \][/tex]
6. Group the terms in pairs and factor out the common factors in each pair:
[tex]\[ 3x(x - 1) - 5(x - 1) \][/tex]
7. Now, factor out the common binomial [tex]\((x - 1)\)[/tex]:
[tex]\[ (x - 1)(3x - 5) \][/tex]
Thus, the correct factorization of the quadratic expression [tex]\(3x^2 - 8x + 5\)[/tex] is:
[tex]\[ (x - 1)(3x - 5) \][/tex]
So, among the given options, the correct choice is:
[tex]\[(3x - 5)(x - 1)\][/tex]
