To factorise x³ + 13x² + 32x + 20, we need to follow a step-by-step process.



The first thing we should do is look for any common factors that we can factor out. In this case, we can factor out a common factor of 2:

x³ + 13x² + 32x + 20 = 2(x³/2 + 13x²/2 + 16x + 10)



Now that we have factored out the common factor of 2, we can focus on factoring the quadratic expression inside the parentheses. To do this, we need to find two numbers that multiply to 10 and add up to 16. These numbers are 2 and 8:

x³ + 13x² + 32x + 20 = 2(x + 2)(x² + 11x + 10)



The quadratic expression x² + 11x + 10 can be factored into (x + 1)(x + 10):

x³ + 13x² + 32x + 20 = 2(x + 2)(x + 1)(x + 10)



To check that our factorisation is correct, we can expand the expression using FOIL:

2(x + 2)(x + 1)(x + 10) = 2(x³ + 13x² + 32x + 20)

This confirms that our factorisation is correct.

Therefore, the fully factorised form of x³ + 13x² + 32x + 20 is 2(x + 2)(x + 1)(x + 10).

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