Question

Factor completely.
\[7x^2+35x+42=\]

Answer 1

To factor the expression \(7x^2 + 35x + 42\), we start by identifying the greatest common factor (GCF) of all the terms in the polynomial.

The coefficients are:
- \(7\) for \(7x^2\)
- \(35\) for \(35x\)
- \(42\) (the constant term)

The GCF of \(7\), \(35\), and \(42\) is \(7\). We can factor \(7\) out of the polynomial:

\[
7(x^2 + 5x + 6)
\]

Next, we will factor the quadratic expression \(x^2 + 5x + 6\). We need to find two numbers that multiply to \(6\) (the constant term) and add to \(5\) (the coefficient of \(x\)). The numbers \(2\) and \(3\) fit this requirement because:

\[
2 \times 3 = 6 \quad \text{and} \quad 2 + 3 = 5
\]

Now, we can factor \(x^2 + 5x + 6\) as:

\[
(x + 2)(x + 3)
\]

Putting it all together, we have:

\[
7(x + 2)(x + 3)
\]

Thus, the completely factored form of the expression \(7x^2 + 35x + 42\) is:

\[
\boxed{7(x + 2)(x + 3)}
\]