Question

Completely factor the expression. $$7 x^{2}+8 x+1$$

Short answer

The factored form of the quadratic expression is \((7x + 1)(x + 1)\).

Step-by-step solution

Identify the terms in the quadratic expression

The quadratic expression given is \(7x^2 + 8x + 1\). Here, a = 7, b = 8, and c = 1.

Find the pairs

We need to find two numbers e and g such that e*g is equal to c (here, c = 1), and their sum multiplied by a (here, 7) gives b (here, 8). The numbers e, g that satisfy these conditions are 1 and 1 (because 1*1 = 1 and 7*(1 + 1) = 8).

Determine d and f

The numbers d and f are both 7 because when d and f are multiplied they give a (which is 7) and it's also stated that d = f since e = g.

Write the factors

Using the values of d, e and f, g, the factored form of the quadratic expression is \((7x + 1)(x + 1)\).