To simplify the expression \( 4x^2 - 3x^2 + 5x - 24 \), we first combine like terms:
\[ 4x^2 - 3x^2 = (4 - 3)x^2 = 1x^2 = x^2 \]
Thus, the expression simplifies to:
\[ x^2 + 5x - 24 \]
Now we will factor \( x^2 + 5x - 24 \) to find its equivalent expression:
We need to factor this quadratic expression. We are looking for two numbers that multiply to \(-24\) (the constant term) and add up to \(5\) (the coefficient of the linear term).
The two numbers that satisfy these conditions are \(8\) and \(-3\) because:
\[ 8 \times (-3) = -24 \] \[ 8 + (-3) = 5 \]
Thus, we can factor the expression as follows:
\[ x^2 + 5x - 24 = (x + 8)(x - 3) \]
Now let's check which of the options given is equivalent to this expression:
A) \(2(x-12)\) - This does not factor to \(x^2 + 5x - 24\).
B) \((x+6)(x-4)\) - This expands to \(x^2 + 2x - 24\), which is not correct.
C) \(4x(x-6)\) - This expands to \(4x^2 - 24x\), which is also not correct.
D) \((x+8)(x-3)\) - This matches our factorization.
Thus, the correct answer is D) \((x+8)(x-3)\).