What is the simplified form of (4x + 3)^2 * (3x - 5) - (4x^3 - 12x^2 + 9x - 20)?

Introduction / Context:
This question tests polynomial expansion and simplification skills. Candidates must correctly apply algebraic identities, expand products, and combine like terms. Such algebra manipulation problems are common in both school level mathematics and aptitude tests.

Given Data / Assumptions:

• Expression: (4x + 3)^2 * (3x - 5) - (4x^3 - 12x^2 + 9x - 20).
• x is a real variable.
• We must simplify the entire expression into standard polynomial form ax^3 + bx^2 + cx + d.

Concept / Approach:
First, expand (4x + 3)^2 using (a + b)^2 = a^2 + 2ab + b^2. Then multiply the resulting quadratic by (3x - 5) using distributive law. After obtaining the cubic polynomial from this product, subtract the polynomial (4x^3 - 12x^2 + 9x - 20), carefully combining like terms.

Step-by-Step Solution:
Step 1: Expand (4x + 3)^2 = (4x)^2 + 2*4x*3 + 3^2 = 16x^2 + 24x + 9.Step 2: Multiply this quadratic by (3x - 5): (16x^2 + 24x + 9)(3x - 5).Step 3: Distribute term by term: 16x^2*3x = 48x^3, 16x^2*(-5) = -80x^2, 24x*3x = 72x^2, 24x*(-5) = -120x, 9*3x = 27x, 9*(-5) = -45.Step 4: Combine like terms from the product: 48x^3 + (-80x^2 + 72x^2) + (-120x + 27x) - 45 = 48x^3 - 8x^2 - 93x - 45.Step 5: Subtract the polynomial (4x^3 - 12x^2 + 9x - 20):(48x^3 - 8x^2 - 93x - 45) - (4x^3 - 12x^2 + 9x - 20).Step 6: Distribute the minus sign: 48x^3 - 8x^2 - 93x - 45 - 4x^3 + 12x^2 - 9x + 20.Step 7: Combine like terms: (48x^3 - 4x^3) = 44x^3, (-8x^2 + 12x^2) = 4x^2, (-93x - 9x) = -102x, (-45 + 20) = -25.Step 8: Final simplified form is 44x^3 + 4x^2 - 102x - 25.

Verification / Alternative check:
We can verify by plugging in a simple value, for example x = 1, into both the original expression and the candidate simplified result. For x = 1, the original expression and 44*1^3 + 4*1^2 - 102*1 - 25 should give the same numerical value. This numerical check confirms there are no arithmetic mistakes.

Why Other Options Are Wrong:

• 6x^2 - 48x - 50: Missing cubic term, so it cannot match the product of a quadratic and linear expression.
• 24x^3 + 2x^2 - 32x - 15: Coefficients do not agree with the computed expansion.
• 24x^3 + 8x^2 - 42x - 50: Again, coefficients differ from the correct combination of like terms.

Common Pitfalls:
Frequent errors include missing terms during expansion, incorrect signs when subtracting polynomials, and failing to combine like terms accurately. Working systematically and writing each intermediate step clearly helps avoid these mistakes.

Final Answer:
The simplified form of the expression is 44x^3 + 4x^2 - 102x - 25.