Question

1. factor. $x^2 + 7x + 12$

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Explanation:

Step1: Find pair summing to 7, product 12

We need two numbers that add up to 7 and multiply to 12. These numbers are 3 and 4, since $3+4=7$ and $3\times4=12$.

Step2: Split middle term

Rewrite the quadratic by splitting the $7x$ term:
$x^2 + 3x + 4x + 12$

Step3: Factor by grouping

Group the first two and last two terms, then factor out common terms:
$(x^2 + 3x) + (4x + 12) = x(x+3) + 4(x+3)$

Step4: Factor out common binomial

Factor out the shared $(x+3)$ term:
$(x+3)(x+4)$

Answer:

$(x+3)(x+4)$