To factor the quadratic expression [tex]\(7x^2 + 35x + 42\)[/tex], follow these steps:
1. Identify the quadratic expression:
[tex]\[ 7x^2 + 35x + 42 \][/tex]
2. Look for common factors:
First, we observe that each term in the quadratic expression has a common factor. For the given expression, 7 is a common factor of all the terms. We factor out 7:
[tex]\[ 7(x^2 + 5x + 6) \][/tex]
3. Factor the quadratic trinomial:
Next, we need to factor the quadratic expression within the parentheses: [tex]\(x^2 + 5x + 6\)[/tex].
To factor [tex]\(x^2 + 5x + 6\)[/tex], we look for two numbers that multiply to 6 (the constant term) and add up to 5 (the coefficient of the linear term). The numbers 2 and 3 fit this requirement because:
[tex]\[ 2 \times 3 = 6 \][/tex]
[tex]\[ 2 + 3 = 5 \][/tex]
4. Rewrite and factor the trinomial:
Using the numbers 2 and 3, we can write the trinomial as a product of two binomials:
[tex]\[ x^2 + 5x + 6 = (x + 2)(x + 3) \][/tex]
5. Combine the factors:
We now place the previously factored out 7 back with our binomials:
[tex]\[ 7(x + 2)(x + 3) \][/tex]
Therefore, the completely factored form of the given quadratic expression [tex]\(7x^2 + 35x + 42\)[/tex] is:
[tex]\[ 7(x + 2)(x + 3) \][/tex]
