Factorize completely $\frac{x^{3} + 3x^{2} - 10x}{2x^{2} - 8}$

Analyze the question: We need to factorize the numerator and the denominator separately, then cancel any common factors.

Numerator: x³ + 3x² − 10x = x(x² + 3x − 10)

To factor x² + 3x − 10, find two numbers that multiply to −10 and add to +3. These are +5 and −2.

x² + 3x − 10 = (x + 5)(x − 2)

So the numerator = x(x + 5)(x − 2)

Denominator: 2x² − 8 = 2(x² − 4) = 2(x − 2)(x + 2), using the difference of two squares.

Simplify:

Compare with the options: The options are written in a garbled way, but the pattern is clear. Option C, "x(x+5)2(x+2)", is meant to be x(x+5) / [2(x+2)], with the fraction bar lost. That matches our result exactly.

A and B have (x − 5), which is wrong because the factor is (x + 5). Option D does not match the factored form.

Tip: In factorization questions, always take out the common factor first (here x and 2). Then factor the quadratic and look for cancellations such as (x − 2). Remember that a² − b² = (a − b)(a + b).