Final answer:
Option A is the correct representation of the given expression as a binomial square because it is the only option that matches the original coefficients when expanded: -4(2x - 3)².
Step-by-step explanation:
The given expression is -32x²+12x+36, which needs to be expressed as a binomial square. To accomplish this, the expression must be in the form of (ax+b)² = a²x² + 2abx + b². Let's find out which of the provided options is the correct binomial square representation of the given expression by expanding each one.
- Option A: -4(2x - 3)² = -4[(2x)² - 2(2x)(3) + (3)²] = -4[4x² - 12x + 9] = -16x² + 48x - 36.
- Option B: (2x - 3)² = [(2x)² - 2(2x)(3) + (3)²] = 4x² - 12x + 9.
- Option C: -2(4x + 3)² = -2[(4x)² + 2(4x)(3) + (3)²] = -2[16x² + 24x + 9] = -32x² - 48x - 18.
- Option D: (4x + 3)² = [(4x)² + 2(4x)(3) + (3)²] = 16x² + 24x + 9.
Comparing each expanded form with the original expression, we see that option A is the correct form because its expansion results in the same coefficients for all terms as the given expression.