Q. \(x^2+7x+12\)

We want to simplify the expression

\(x^2 + 7x + 12\).

The expression is a quadratic, so we look for a way to factor it.

Step 1: Identify the form

A quadratic can often be written in factored form as

\((x+a)(x+b)\).

When we expand \((x+a)(x+b)\), we get

\((x+a)(x+b)=x^2+(a+b)x+ab\).

So we match coefficients with \(x^2 + 7x + 12\).

Step 2: Match the coefficients

We need:

\(a+b=7\)

and

\(ab=12\).

Step 3: Find numbers that satisfy both conditions

We need two numbers whose product is \(12\) and whose sum is \(7\).

The pair \(3\) and \(4\) works because:

\(3+4=7\)

and

\(3\cdot 4=12\).

Step 4: Write the factored form

Substitute \(a=3\) and \(b=4\) into \((x+a)(x+b)\):

\((x+3)(x+4)\).

Final Answer

\(x^2 + 7x + 12 = (x+3)(x+4)\).