factor 𝑥(𝑥^2+4𝑥−12) / 16𝑥^2−9
To factor the expression 𝑥(𝑥^2 + 4𝑥 − 12) / (16𝑥^2 − 9), we start by factoring the numerator and denominator separately.
The numerator can be factored as 𝑥(𝑥 + 6)(𝑥 − 2).
The denominator is a difference of squares, which can be factored as (4𝑥 − 3)(4𝑥 + 3).
Therefore, the entire expression can be factored as 𝑥(𝑥 + 6)(𝑥 − 2) / (4𝑥 − 3)(4𝑥 + 3).
Note: It's important to mention that we should always check if any of the factors we obtained make the denominator equal to zero, as that would make the expression undefined. In this case, the factor 4𝑥 − 3 would make the denominator equal to zero, so 𝑥 ≠ 3/4. Similarly, the factor 4𝑥 + 3 would also make the denominator equal to zero, so 𝑥 ≠ -3/4.