Factor the trinomial:5, x, squared, plus, 22, x, plus, 215x 2 +22x+21

Sure, let's factor the trinomial step by step.

The trinomial is 5x^2 + 22x + 21.

Step 1: Identify a, b, and c in the trinomial ax^2 + bx + c. Here, a = 5, b = 22, and c = 21.

Step 2: Find two numbers that multiply to give ac (a times c) and add to give b. So we need two numbers that multiply to 521 = 105 and add to 22. These numbers are 15 and 7 because 157 = 105 and 15 + 7 = 22.

Step 3: Rewrite the trinomial as a four-term polynomial by splitting the bx term using the two numbers found in step 2. So, 5x^2 + 22x + 21 becomes 5x^2 + 15x + 7x + 21.

Step 4: Factor by grouping. The first two terms 5x^2 and 15x have a common factor of 5x, and the last two terms 7x and 21 have a common factor of 7. So, the expression becomes 5x(x + 3) + 7(x + 3).

Step 5: Notice that (x + 3) is a common factor of both terms. So, the factored form of the trinomial is (5x + 7)(x + 3).

So, the trinomial 5x^2 + 22x + 21 factors to (5x + 7)(x + 3).