Question

How do you solve `(x + 4) ^ { 3} - 8= x ( x + 5) ( x + 7) - 5`?

Answer 1

Multiply out and cancel down - most of the difficulty immediately disappears in this particular problem

Explanation:

Multiply it all out, recalling the binomial theorem:
`x^3+12x^2+48x+64-8=x^3+12x^2+35x-5`
Collect terms, noticing that the cubic and quadratic terms cancel:
`13x+61=0`

So `x=-61/13=-4.69` to three significant figures, a very much simpler answer than might have been suspected from a quick inspection of the initial equation.

Plotting the two sides of the equation to sanity check it isn't easily done by hand, but it's still nice to see them crossing only once:
graph{(y-((x+4)^3-8))(y-(x(x+5)(x+7)-5))=0 [-10, 5, -200, 500]}