To simplify the polynomial expression [tex]\( 3x(4x + 5) - 4(-x - 3)(2x - 5) \)[/tex], we can follow a step-by-step approach.
Step 1: Distribute within the first term:
[tex]\[ 3x(4x + 5) = 3x \cdot 4x + 3x \cdot 5 = 12x^2 + 15x \][/tex]
Step 2: Distribute within the second term:
[tex]\[ -4(-x - 3)(2x - 5) \][/tex]
First, distribute [tex]\(-4\)[/tex]:
[tex]\[ -4(-x - 3)(2x - 5) = -4 \left[ (-x)(2x) + (-x)(-5) + (-3)(2x) + (-3)(-5) \right] \][/tex]
Simplify each product:
[tex]\[ (-x)(2x) = -2x^2 \][/tex]
[tex]\[ (-x)(-5) = 5x \][/tex]
[tex]\[ (-3)(2x) = -6x \][/tex]
[tex]\[ (-3)(-5) = 15 \][/tex]
Step 3: Combine the terms from the second term:
[tex]\[ -4(-x - 3)(2x - 5) = -4(-2x^2 + 5x - 6x + 15) \][/tex]
Simplify the expression within the parentheses:
[tex]\[ -2x^2 + 5x - 6x + 15 = -2x^2 - x + 15 \][/tex]
Distribute the [tex]\(-4\)[/tex] to each term:
[tex]\[ -4(-2x^2 - x + 15) = 8x^2 + 4x - 60 \][/tex]
Step 4: Combine the results from both steps:
From Step 1: [tex]\( 12x^2 + 15x \)[/tex]
From Step 2: [tex]\( + 8x^2 + 4x - 60 \)[/tex]
[tex]\[ 12x^2 + 15x + 8x^2 + 4x - 60 \][/tex]
Step 5: Combine like terms:
[tex]\[ (12x^2 + 8x^2) + (15x + 4x) - 60 \][/tex]
[tex]\[ 20x^2 + 19x - 60 \][/tex]
Thus, the simplified form of the polynomial expression [tex]\( 3x(4x + 5) - 4(-x - 3)(2x - 5) \)[/tex] is:
[tex]\[ 20x^2 + 19x - 60 \][/tex]
The correct answer is [tex]\( 20x^2 + 19x - 60 \)[/tex].
