Question 21

The number of values $$a$$ can take such that $$x^4 + ax^3 + (3a - 4)x^2 + 2(a - 1)x - 4$$ can be expressed as a product of two quadratic polynomials, $$x^2 + px + 2$$ and $$x^2 + qx - 2$$, where $$p$$ and $$q$$ are real, is

Expand $$(x^2 + px + 2)(x^2 + qx - 2) = x^4 + (p+q)x^3 + pq \cdot x^2 + 2(q-p)x - 4$$.

Comparing with $$x^4 + ax^3 + (3a-4)x^2 + 2(a-1)x - 4$$:

Solving: $$q = \dfrac{2a-1}{2}$$, $$p = \dfrac{1}{2}$$. Substituting into $$pq = 3a-4$$: $$\dfrac{2a-1}{4} = 3a - 4 \Rightarrow a = \dfrac{3}{2}$$.

Only 1 value of $$a$$ works.

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