How can we make this a perfect square?

Question

The question I have is: How can this be transformed into a perfect square?

$$a(a+1)(a+2)(a+3)+1$$

Answer 1

If you expand it out, you notice that

$$a(a+1)(a+2)(a+3) + 1 = a^4 + 6 a^3 + 11 a^2 + 6 a + 1$$

If this is to be a perfect square, it is likely in the form

$$a^4 + 6 a^3 + 11 a^2 + 6 a + 1 = (a^2 + \gamma a + 1)^2 \tag 1$$

based on:

• the fact it has five terms
• the fact that it has $a^4$ (so the square root is $a^2$)
• the constant term is $1$

The issue being what $\gamma$ is. Fully expanding this

$$(a^2 + \gamma a + 1)^2 = a^4 + 2 a^3 \gamma + a^2 \gamma^2 + 2 a^2 + 2 a \gamma + 1 \tag 2$$

Comparing the coefficients in $(1)$ and $(2)$ we see that, clearly,

$$2 \gamma = 6 \qquad 2 + \gamma^2 = 11 \qquad 2\gamma = 6$$

Clearly, then, $\gamma = 3$.