Factoring help

Question

Show step by step why/how the left side equals the right side.

Answer 1

The equation is 1 - 4 ln(x) - ln²(x) = 5 - (ln x + 2)².

The left hand side expression is 1 - 4 ln(x) - ln²(x).

Let ln(x) = t ⇒ 1 - 4t - t².

Write the equation in complete the square form.

x-coefficient = - 4 and add and subtract (x-coefficient/2)² = (-4/2)² = 4 to the expression.

= (1 - 4t - t²) + 4 - 4

= (- 4 - 4t - t²) + 4 + 1

= 5 - (t + 2)²

= 5 - (ln x + 2)²

= right hand side expression.