Sure! Let's simplify the expression \(\frac{5x^2 - 80}{x^2 - 7x + 12}\) step-by-step.
Step 1: Factor the Numerator
The numerator is \(5x^2 - 80\).
First, factor out the common factor in the numerator: \(5\).
[tex]\[ 5x^2 - 80 = 5(x^2 - 16) \][/tex]
Next, notice that \(x^2 - 16\) is a difference of squares. Recall the difference of squares formula: \(a^2 - b^2 = (a + b)(a - b)\).
Apply this to \(x^2 - 16\):
[tex]\[ x^2 - 16 = (x + 4)(x - 4) \][/tex]
Therefore, the numerator can be factored as:
[tex]\[ 5(x^2 - 16) = 5(x + 4)(x - 4) \][/tex]
Step 2: Factor the Denominator
The denominator is \(x^2 - 7x + 12\).
We need to factor this quadratic expression. We look for two numbers that multiply to \(12\) and add up to \(-7\). These numbers are \(-3\) and \(-4\).
Thus, we can write the quadratic expression as a product of two binomials:
[tex]\[ x^2 - 7x + 12 = (x - 3)(x - 4) \][/tex]
Step 3: Form the Fraction with Factored Terms
Now, rewrite the expression with the factored numerator and denominator:
[tex]\[ \frac{5x^2 - 80}{x^2 - 7x + 12} = \frac{5(x + 4)(x - 4)}{(x - 3)(x - 4)} \][/tex]
Step 4: Simplify the Expression
Notice that \((x - 4)\) appears in both the numerator and the denominator, so they can be cancelled out:
[tex]\[ \frac{5(x + 4)(x - 4)}{(x - 3)(x - 4)} = \frac{5(x + 4) \cancel{(x - 4)}}{(x - 3) \cancel{(x - 4)}} \][/tex]
After canceling, we are left with:
[tex]\[ \frac{5(x + 4)}{x - 3} \][/tex]
Thus, the simplified form of the expression is:
[tex]\[ \boxed{\frac{5(x + 4)}{x - 3}} \][/tex]
